A036039 -id:A036039 - cp's OEIS frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 3, 6, 1, 4, 6, 5, 1, 1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1, 1, 2, 2, 2, 1, 3, 6, 6, 3, 3, 4, 12, 6, 12, 1, 5, 20, 10, 6, 15, 7, 1, 1, 2, 2, 2, 2, 3, 6, 6, 3, 3, 6, 1, 4, 12, 12, 12, 12, 4, 5, 20, 10, 30, 5, 6, 30, 20, 7, 21, 8, 1

Offset: 0

Alford Arnold

This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).
For a signed version see A111786.
These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021

			Table starts:
[1]
[1]
[1, 1]
[1, 2, 1]
[1, 2, 1, 3, 1]
[1, 2, 2, 3, 3, 4, 1]
[1, 2, 2, 1, 3, 6, 1, 4, 6,  5, 1]
[1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1]
.
T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210
T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - _Geoffrey Critzer_, May 19 2013
These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of [5] :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - _Wolfdieter Lang_, May 30 2018

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972.
Wolfdieter Lang, First 10 rows and more.

Cf. A000670, A007318, A036035, A036038, A019538, A115621, A309004, A000079 (row sums), A000041 (row lengths).

nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036040(n, m) := (add(q(t), t=1..n))!/(mul(q(t)!, t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

C(sig)={my(S=Set(sig)); (#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A048996_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [factorial(len(p))//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019 [corrected on notice from Sean A. Irvine, Apr 30 2022]

T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k).
If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006
T(n,k) = A309004(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

a(0)=1 prepended by Andrew Howroyd, Oct 19 2020

A271703 Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 12, 1, 0, 120, 240, 120, 20, 1, 0, 720, 1800, 1200, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1

Offset: 0

Peter Luschny, Apr 14 2016

The Lah numbers can be seen as the case m=1 of the family of triangles T_{m}(n,k) = T_{m}(n-1,k-1)+(k^m+(n-1)^m)*T_{m}(n-1,k) (see the link 'Partition transform').

This is the Sheffer triangle (lower triangular infinite matrix) (1, x/(1-x)), an element of the Jabotinsky subgroup of the Sheffer group. - Wolfdieter Lang, Jun 12 2017

			As a rectangular array (diagonals of the triangle):
  1,      1,       1,       1,       1,       1,       ... A000012
  0,      2,       6,       12,      20,      30,      ... A002378
  0,      6,       36,      120,     300,     630,     ... A083374
  0,      24,      240,     1200,    4200,    11760,   ... A253285
  0,      120,     1800,    12600,   58800,   211680,  ...
  0,      720,     15120,   141120,  846720,  3810240, ...
A000007, A000142, A001286, A001754, A001755,  A001777.
The triangle T(n,k) begins:
n\k 0       1        2        3        4       5      6     7    8  9 10 ...
0:  1
1:  0       1
2:  0       2        1
3:  0       6        6        1
4:  0      24       36       12        1
5:  0     120      240      120       20       1
6:  0     720     1800     1200      300      30      1
7:  0    5040    15120    12600     4200     630     42     1
8:  0   40320   141120   141120    58800   11760   1176    56    1
9:  0  362880  1451520  1693440   846720  211680  28224  2016   72  1
10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90  1
...  - _Wolfdieter Lang_, Jun 12 2017

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.
I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.
T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
M. F. Hasler and Peter Luschny, Formulas for A271703, OEIS Wiki, Aug. 2017.
S. A. Joni, G.-C. Rota, and B. Sagan, From sets to functions: Three elementary examples, Discrete Mathematics, Volume 37, Issues 2-3, 1981, 193-202.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, Lah numbers
Peter Luschny, Partition transform
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 18.
Piotr Miska and Maciej Ulas, On some properties of the number of permutations being products of pairwise disjoint d-cycles, arXiv:1904.03395 [math.NT], 2019.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Variants: A008297 the main entry for these numbers, A105278, A111596 (signed).
A000262 (row sums). Largest number of the n-th row in A002868.
Cf. A097805, A354795, A360177.

T := (n, k) -> `if`(n=k, 1, binomial(n-1,k-1)*n!/k!):
seq(seq(T(n, k), k=0..n), n=0..9);

T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

@cached_function
def T(n,k):
    if k<0 : return 0
    if k==n: return 1
    return T(n-1,k-1) + (k+n-1)*T(n-1,k)
for n in (0..8): print([T(n,k) for k in (0..n)])

For a collection of formulas see the 'Lah numbers' link.
T(n, k) = A097805(n, k)*n!/k! = (-1)^k*P_{n, k}(1,1,1,...) where P_{n, k}(s) is the partition transform of s.
T(n, k) = coeff(n! * P(n), x, k) with P(n) = (1/n)*(Sum_{k=0..n-1}(x(n-k)*P(k))), for n >= 1 and P(n=0) = 1, with x(n) = n*x. See A036039. - Johannes W. Meijer, Jul 08 2016
From Wolfdieter Lang, Jun 12 2017: (Start)
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (that is egf of the triangle) is exp(x*t/(1-t)) (a Sheffer triangle of the Jabotinsky type).
E.g.f. column k: (t/(1-t))^k/k!.
Three term recurrence: T(n, k) = T(n-1, k-1) + (n-1+k)*T(n, k-1), n >= 1, k = 0..n, with T(0, 0) =1, T(n, -1) = 0, T(n, k) = 0 if n < k.
T(n, k) = binomial(n, k)*fallfac(x=n-1, n-k), with fallfac(x, n) = Product_{j=0..(n-1)} (x - j), for n >= 1, and 0 for n = 0.
risefac(x, n) = Sum_{k=0..n} T(n, k)*fallfac(k), with risefac(x, n) = Product_{j=0..(n-1)} (x + j), for n >= 1, and 0 for n = 0.
See Graham et al., exercise 31, p. 312, solution p. 552. (End)
Formally, let f_n(x) = Sum_{k>n} (k-1)!*x^k; then f_n(x) = Sum_{k=0..n} T(n, k)* x^(n+k)*f_0^((k))(x), where ^((k)) stands for the k-th derivative. - Luc Rousseau, Dec 27 2020
T(n, k) = Sum_{j=k..n} A354795(n, j)*A360177(j, k). - Mélika Tebni, Feb 02 2023
T(n, k) = binomial(n, k)*(n-1)!/(k-1)! for n, k > 0. - Chai Wah Wu, Nov 30 2023

A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 6, 5, 4, 1, 1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1, 1, 6, 14, 14, 15, 35, 21, 21, 20, 35, 14, 15, 14, 6, 1, 1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, 28, 21, 20, 7, 1

Offset: 0

Wolfdieter Lang, Apr 13 2006

The n-th row has partition(n) = A000041(n) entries.
Also the numbers of standard Young tableaux for Young diagrams (or partitions).
Also "generalized" Catalan numbers. For a partition of n, n=(n_1+...+n_d), this is the number of integral lattice paths from (0,...,0) to (n_1,...,n_d) such that for any point p=(p_1,...p_d) on such a path p_i is never less than p_j whenever iGraham H. Hawkes, Jul 05 2013
The irreducible representations of S_n correspond to Young diagrams or partitions.
Partitions of n are ordered according to Abramowitz-Stegun (A-St) (see the reference, pp. 831-2). In contrast to A-St, a partition has nondecreasing parts (reverse notation of A-St).
The dimension of a representation of S_n corresponding to a Young diagram or partition is a(n,k) for the k-th partition of n in this A-St order.
One could call these numbers a(n,k) M_4 (similar to M_0, M_1, M_2, M_3 given in A111786, A036038, A036039, A036040, respectively).
From Wolfdieter Lang, Oct 09 2015: (Start)
The first formula given below appears in A. Young, Q.S.A. III, PLMS 28 (1928) 255-292 (third paper on "On Quantitative Substitutional Analysis"), Theorem II on p. 260, and he calls it f; see the collected papers (CP) reference, p. 357. Note the shorthand notation for the products; see Q.S.A. II, PLMS 34 (1902) 361-397, p. 366, CP, p. 97, for the explicit one.
This formula also can be found in the Glass-Ng link, Theorem 1, p. 702, using the Vandermonde determinant in the numerator and re-indexing the denominator.
The product of the hook length numbers, called H(n, k) in this formula below, is found in A263003(n, k).
The squared row entries sum to n!. See A. Young, Q.S.A. II (see above), pp. 367-368, CP pp. 98-99. Also Q.S.A. III, p. 265, CP p. 362.
(End)

			[1];
[1];
[1, 1];
[1, 2, 1];
[1, 3, 2, 3, 1];
[1, 4, 5, 6, 5, 4, 1];
[1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1];...
a(4,4)=3 because the 4th partition of n=4 in A-St order is [2,1,1],
and H(4,4)=(4!*2!*1!)/Vandermonde([4,2,1]) = (4!*2)/6 =4*2, hence
4!/H(4,4) = 3.
a(4,4)=3 because the hook lengths of the Young diagram of [2,1,1] are [4, 1; 2; 1], hence 4!/(4*1*2*1) = 3.
The sum of the squared entries of each row gives n!: n = 5: 2*(1^1 + 4^2 + 5^2) + 6^2 = 120 = 5!. - _Wolfdieter Lang_, Oct 09 2015

G. de B. Robinson (ed.), The Collected Papers of Alfred Young 1873-1940, University of Toronto Press, 1977.
G. B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

Alois P. Heinz, Rows n = 0..30, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, pp. 831-2.
Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
Graham H. Hawkes, An Elementary Proof of a Formula for SYT, arXiv preprint arXiv:1310.5919 [math.CO], 2013-2014.
Wolfdieter Lang, First 15 rows.
Eric Weisstein's World of Mathematics, Hook length formula.
Doron Zeilberger, Andre's Reflection Proof Generalized to the Many-Candidate Ballot Problem, Discrete Mathematics 44 (1983) 325-326.
Index entries for sequences related to groups

Cf. A000041, A000085 (row sums), A060240 (rows sorted), A263003.

Cf. A067924.

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],
    [g(n, i-1, l)[], g(n-i, min(n-i, i), [l[], i])[]]):
T:= n-> map(x-> n!/x, g(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015

h[l_List] := Function[n, Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]]; g[n_, i_, l_List] := If[n==0 || i==1, Join[{h[Join[l, Array[1&, n]]]}], If[i<1, {}, Join[{g[n, i-1, l]}, If[i>n, {}, g[n-i, i, Join[l, {i}]]]]]] // Flatten; T[n_] := n!/ g[n, n, {}]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)

a(n,k) = n!/H(n,k) with H(n,k):= Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k)-j)) with the Vandermonde determinant for the variables x_i:=lambda(n,k)_i + m(n,k)-i, i,j=1..m(n,k) if m(n,k) is the number of parts of the k-th partition of n, called lambda(n,k), in the A-St order (see above). Lambda(n,k)_i denotes the i-th part of the partition lambda(n,k), sorted in decreasing order (this is the reverse of the A-St notation).

a(n,k) = n!/Product_{j=1..n}(h(n,k,j) with the hook numbers h(n,k,j) of the Young diagram of the partition lambda(n,k) in the A-St order. See the link for 'hook length formula'.

Row n=0 prepended by Alois P. Heinz, Nov 05 2015

A263916 Coefficients of the Faber partition polynomials.

Original entry on oeis.org

-1, -2, 1, -3, 3, -1, -4, 4, 2, -4, 1, -5, 5, 5, -5, -5, 5, -1, -6, 6, 6, -6, 3, -12, 6, -2, 9, -6, 1, -7, 7, 7, -7, 7, -14, 7, -7, -7, 21, -7, 7, -14, 7, -1, -8, 8, 8, -8, 8, -16, 8, 4, -16, -8, 24, -8, -8, 12, 24, -32, 8, 2, -16, 20, -8, 1

Offset: 1

Tom Copeland, Oct 29 2015

The coefficients of the Faber polynomials F(n,b(1),b(2),...,b(n)) (Bouali, p. 52) in the order of the partitions of Abramowitz and Stegun. Compare with A115131 and A210258.
These polynomials occur in discussions of the Virasoro algebra, univalent function spaces and the Schwarzian derivative, symmetric functions, and free probability theory. They are intimately related to symmetric functions, free probability, and Appell sequences through the raising operator R = x - d log(H(D))/dD for the Appell sequence inverse pair associated to the e.g.f.s H(t)e^(xt) (cf. A094587) and (1/H(t))e^(xt) with H(0)=1.
Instances of the Faber polynomials occur in discussions of modular invariants and modular functions in the papers by Asai, Kaneko, and Ninomiya, by Ono and Rolen, and by Zagier. - Tom Copeland, Aug 13 2019
The Faber polynomials, denoted by s_n(a(t)) where a(t) is a formal power series defined by a product formula, are implicitly defined by equation 13.4 on p. 62 of Hazewinkel so as to extract the power sums of the reciprocals of the zeros of a(t). This is the Newton identity expressing the power sum symmetric polynomials in terms of the elementary symmetric polynomials/functions. - Tom Copeland, Jun 06 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			F(1,b1) = - b1
F(2,b1,b2) = -2 b2 + b1^2
F(3,b1,b2,b3) = -3 b3 + 3 b1 b2 - b1^3
F(4,b1,...) = -4 b4 + 4 b1 b3 + 2 b2^2  - 4 b1^2 b2 + b1^4
F(5,...) = -5 b5 + 5 b1 b4 + 5 b2 b3 - 5 b1^2 b3 - 5 b1 b2^2 + 5 b1^3 b2 - b1^5
------------------------------
IF(1,b1) = -b1
IF(2,b1,,b2) = -b2 + b1^2
IF(3,b1,b2,b3) = -2 b3 + 3 b1 b2 - b1^3
IF(4,b1,...) = -6 b4 + 8 b1 b3 + 3 b2^2  - 6 b1^2 b2 + b1^4
IF(5,...) = -24 b5 + 30 b1 b4 + 20 b2 b3 - 20 b1^2 b3 - 15 b1 b2^2 + 10 b1^3 b2 - b1^5
------------------------------
For 1/(1+x)^2 = 1- 2x + 3x^2 - 4x^3 + 5x^4 - ..., F(n,-2,3,-4,...) = (-1)^(n+1) 2.
------------------------------
F(n,x,2x,...,nx), F(n,-x,2x,-3x,...,(-1)^n n*x), and F(n,(2-x),1,0,0,...) are related to the Chebyshev polynomials through A127677 and A111125. See also A110162, A156308, A208513, A217476, and A220668.
------------------------------
For b1 = p, b2 = q, and all other indeterminates 0, see A113279 and A034807.
For b1 = -y, b2 = 1 and all other indeterminates 0, see A127672.

H. Airault, "Symmetric sums associated to the factorization of Grunsky coefficients," in Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes: Vol. 47, edited by J. Harnad and P. Winternitz, American Mathematical Society, 2009.
D. Bleeker and B. Booss, Index Theory with Applications to Mathematics and Physics, International Press, 2013, (see section 16.7 Characteristic Classes and Curvature).
M. Hazewinkel, Formal Groups and Applications, Academic Press, New York San Francisco London, 1978, p. 120.
F. Hirzebruch, Topological methods in algebraic geometry. Second, corrected printing of the third edition. Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag, Berlin Heidelberg New York, 1978, p. 11 and 92.
D. Knutson, λ-Rings and the Representation Theory of the Symmetric Group, Lect. Notes in Math. 308, Springer-Verlag, 1973, p. 35.
D. Yau, Lambda-Rings, World Scientific Publishing Co., Singapore, 2010, p. 45.

Vincenzo Librandi, Table of n, a(n) for n = 1..5762
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. Airault, Remarks on Faber polynomials, International Mathematical Forum, 3, no. 9, 2008, pages 449-456.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.
H. Airault and Y. Neretin, On the action of the Virasoro algebra on the space of univalent functions, arXiv:0704.2149 [math.RT], 2007.
F. Ardila, Algebraic and geometric methods in enumerative combinatorics, arXiv:1409.2562 [math.CO], 2015.
T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an application, 1997.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, pages 49-70.
J. M. Campbell, Combinatorial interpretations of primitivity in the algebra of symmetric functions, Sarajevo J. Math., 17(30) (2021), 151--165.
P. Cartier, Mathemagics: A tribute to L. Euler and R. Feynman, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d, 2000, p. 53.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
V. Chan, Topological K-theory of complex projective spaces, senior's thesis UC Davis (p. 11 on Chern characters of complex vector bundles), 2013.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, An algebraic structure for Faber polynomials, Lin. Alg. Applic. 433 (2010) 1170-1179.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, MathOverflow question, 2012.
Tom Copeland, Addendum to Elliptic Lie Triad, 2015.
Tom Copeland, Connection between the Chebyshev polynomials and the Faber polynomials, MathOverflow question, 2015.
Tom Copeland, The Faber Appells, 2020.
Tom Copeland, Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
A. Dress and C. Siebeneicher, The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, lambda-rings, and the universal ring of Witt vectors, Adv. in Math., Vol. 78, Issue 1, Nov. 1989, pages 1-41.
D. Dugger, A Geometric Introduction to K-Theory, p. 288 (called Newton polynomials).
M. Eiermann and R. Varga, Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains, Elect. Trans. on Numer. Analysis, Vol. 1, p. 49-71, 1993.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (normalized versions on pp. 42 and 78, denoted as Schur polynomials).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
A. Hatcher, Vector Bundles and K-Theory, Version 2.2, 2017, p. 63.
M. Hazewinkel, Three lectures on formal group laws, Canadian Mathematical Society Conference Proceedings, Vol. 5, p. 51-67, 1986.
Y. He, The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning, arXiv:1812.02893 [hep-th], 2018, p. 146.
O. Iena, On symbolic computations with Chern classes: Remarks on the library CHERN.LIB for SINGULAR, p. 6.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
R. Lu, Regularized equivariant Euler classes and gamma functions, Ph.D. thesis, Dept. of Pure Mathematics, University of Adelaide, 2008, p. 86.
K. Maslanka, Effective method of computing Li’S coefficients and their propertie, arXiv:0402168v5 [math.NT], 2004, p. 6.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
J. McKay and A. Sebbar, On replicable functions: an introduction, Frontiers in Number Theory, Physics, and Geometry II, pp. 373-386.
L. Nicolaescu, Lectures on the Geometry of Manifolds, p.337, 2018.
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
K. Ono and L. Rolen, On Witten's extremal partition functions , arXiv:1807.00444 [math.NT], 2019.
T. Takebe, Lee-Peng Teo, and A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.
Lee-Peng Teo, Analytic functions and integrable hierarchies-characterization of tau functions, Letters in Mathematical Physics, Vol. 64, Issue 1, Apr 2003, pp. 75-92 (also arXiv:hep-th/0305005, 2003).
Wikipedia, Newton identities.
D. Zagier, Traces of singular moduli, 2011.

Cf. Row sums of absolute values: A000225.
Cf. A034807, A036039, A036040, A094587, A110162, A111125, A113279, A115131.
Cf. A127671, A127672, A127677, A130561, A133314, A156308, A208513, A210258.
Cf. A217476, A220668.
Cf. A263634.

F[0] = 1; F[1] = -b[1]; F[2] = b[1]^2 - 2 b[2]; F[n_] := F[n] = -b[1] F[n - 1] - Sum[b[n - k] F[k], {k, 1, n - 2}] - n b[n] // Expand;
row[n_] := (List @@ F[n]) /. b[_] -> 1 // Reverse;
Table[row[n], {n, 1, 8}] // Flatten // Rest (* Jean-François Alcover, Jun 12 2017 *)

-log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n.
-d(1 + b(1) x + b(2) x^2 + ...)/dx / (1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) x^(n-1).
F(n,b(1),...,b(n)) = -n*b(n) - Sum_{k=1..n-1} b(n-k)*F(k,b(1),...,b(k)).
Umbrally, with B(x) = 1 + b(1) x + b(2) x^2 + ..., B(x) = exp[log(1-F.x)] and 1/B(x) = exp[-log(1-F.x)], establishing a connection to the e.g.f. of A036039 and the symmetric polynomials.
The Stirling partition polynomials of the first kind St1(n,b1,b2,...,bn;-1) = IF(n,b1,b2,...,bn) (cf. the Copeland link Lagrange a la Lah, signed A036039, and p. 184 of Airault and Bouali), i.e., the cyclic partition polynomials for the symmetric groups, and the Faber polynomials form an inverse pair for isolating the indeterminates in their definition, for example, F(3,IF(1,b1),IF(2,b1,b2)/2!,IF(3,b1,b2,b3)/3!)= b3, with bk = b(k), and IF(3,F(1,b1),F(2,b1,b2),F(3,b1,b2,b3))/3!= b3.
The polynomials specialize to F(n,t,t,...) = (1-t)^n - 1.
See Newton Identities on Wikipedia on relation between the power sum symmetric polynomials and the complete homogeneous and elementary symmetric polynomials for an expression in multinomials for the coefficients of the Faber polynomials.
(n-1)! F(n,x[1],x[2]/2!,...,x[n]/n!) = - p_n(x[1],...,x[n]), where p_n are the cumulants of A127671 expressed in terms of the moments x[n]. - Tom Copeland, Nov 17 2015
-(n-1)! F(n,B(1,x[1]),B(2,x[1],x[2])/2!,...,B(n,x[1],...,x[n])/n!) = x[n] provides an extraction of the indeterminates of the complete Bell partition polynomials B(n,x[1],...,x[n]) of A036040. Conversely, IF(n,-x[1],-x[2],-x[3]/2!,...,-x[n]/(n-1)!) = B(n,x[1],...,x[n]). - Tom Copeland, Nov 29 2015
For a square matrix M, determinant(I - x M) = exp[-Sum_{k>0} (trace(M^k) x^k / k)] = Sum_{n>0} [ P_n(-trace(M),-trace(M^2),...,-trace(M^n)) x^n/n! ] = 1 + Sum_{n>0} (d[n] x^n), where P_n(x[1],...,x[n]) are the cycle index partition polynomials of A036039 and d[n] = P_n(-trace(M),-trace(M^2),...,-trace(M^n)) / n!. Umbrally, det(I - x M)= exp[log(1 - b. x)] = exp[P.(-b_1,..,-b_n)x] = 1 / (1-d.x), where b_k = tr(M^k). Then F(n,d[1],...,d[n]) = tr[M^n]. - Tom Copeland, Dec 04 2015
Given f(x) = -log(g(x)) = -log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n, action on u_n = F(n,b(1),...,b(n)) with A133932 gives the compositional inverse finv(x) of f(x), with F(1,b(1)) not equal to zero, and f(g(finv(x))) = f(e^(-x)). Note also that exp(f(x)) = 1 / g(x) = exp[Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n] implies relations among A036040, A133314, A036039, and the Faber polynomials. - Tom Copeland, Dec 16 2015
The Dress and Siebeneicher paper gives combinatorial interpretations and various relations that the Faber polynomials must satisfy for integral values of its arguments. E.g., Eqn. (1.2) p. 2 implies [2 * F(1,-1) + F(2,-1,b2) + F(4,-1,b2,b3,b4)]  mod(4) = 0. This equation implies that [F(n,b1,b2,...,bn)-(-b1)^n] mod(n) = 0 for n prime. - Tom Copeland, Feb 01 2016
With the elementary Schur polynomials S(n,a_1,a_2,...,a_n) = Lah(n,a_1,a_2,...,a_n) / n!, where Lah(n,...) are the refined Lah polynomials of A130561, F(n,S(1,a_1),S(2,a_1,a_2),...,S(n,a_1,...,a_n)) = -n * a_n since sum_{n > 0} a_n x^n = log[sum{n >= 0} S(n,a_1,...,a_n) x^n]. Conversely, S(n,-F(1,a_1),-F(2,a_1,a_2)/2,...,-F(n,a_1,...,a_n)/n) = a_n. - Tom Copeland, Sep 07 2016
See Corollary 3.1.3 on p. 38 of Ardila and Copeland's two MathOverflow links to relate the Faber polynomials, with arguments being the signed elementary symmetric polynomials, to the logarithm of determinants, traces of powers of an adjacency matrix, and number of walks on graphs. - Tom Copeland, Jan 02 2017
The umbral inverse polynomials IF appear on p. 19 of Konopelchenko as partial differential operators. - Tom Copeland, Nov 19 2018

More terms from Jean-François Alcover, Jun 12 2017

A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

Original entry on oeis.org

1, 6, 6, 1, 6, 8, 7, 9, 4, 9, 6, 3, 3, 5, 9, 4, 1, 2, 1, 2, 9, 5, 8, 1, 8, 9, 2, 2, 7, 4, 9, 9, 5, 0, 7, 4, 9, 9, 6, 4, 4, 1, 8, 6, 3, 5, 0, 2, 5, 0, 6, 8, 2, 0, 8, 1, 8, 9, 7, 1, 1, 1, 6, 8, 0, 2, 5, 6, 0, 9, 0, 2, 9, 8, 2, 6, 3, 8, 3, 7, 2, 7, 9, 0, 8, 3, 6, 9, 1, 7, 6, 4, 1, 1, 4, 6, 1, 1, 6, 7, 1, 5, 5, 2, 8

Offset: 1

Michael Somos, Sep 02 2005

From Johannes W. Meijer, Jun 27 2016: (Start)
With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.
For similar formulas, see A090998 and A135002. For background information, see A274181.
The structure of the n! * LP(n) formulas leads to the multinomial coefficients A036039. (End)

			1.6616879496335941212958189227499507499644186350250682081897111680...

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

Robert G. Wilson v, Table of n, a(n) for n = 1..1011
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, Section 6.10.
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
M. D. Hirschhorn, A note on Somos' quadratic recurrence constant, J. Number Theory 131 (2011), 2061-2063.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Cristinel Mortici, Estimating the Somos' quadratic recurrence constant, J. Number Theory 130 (2010), 2650-1657.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Wikipedia, Somos' quadratic recurrence constant
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

Cf. A052129, A055209, A116603, A123851, A123852, A123853, A123854.
Cf. A114124 (log).
Cf. A036039, A090998, A135002, A188834, A259235, A274181.

RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2010 *)
Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Apr 07 2014, after Jonathan Sondow *)

{a(n) = if( n<-1, 0, n++; default( realprecision, n+2); floor( prodinf( k=1, k^2^-k)* 10^n) % 10)};

prodinf(n=1,n^2^-n) \\ Charles R Greathouse IV, Apr 07 2013

from mpmath import polylog, diff, exp, mp
mp.dps = 120
somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))
A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Nov 23 2024

Equals Product_{n>=1} n^(1/2^n). - Jonathan Sondow, Apr 07 2013
Equals exp(A114124) = A188834/2 = sqrt(A259235). - Hugo Pfoertner, Nov 23 2024
From Jwalin Bhatt, Apr 02 2025: (Start)
Equals exp(-PolyLog'(0,1/2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.
Equals Product_{n>=1} (1+1/n)^(1/2^n).
Equals exp(Sum_{n>=2} log(n)/2^n).
Equals 2*exp(Sum_{n>=1} (log(1+1/n)-1/n)/2^n). (End)

A130561 Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278; elementary Schur polynomials / functions.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 24, 12, 12, 1, 120, 120, 120, 60, 60, 20, 1, 720, 720, 720, 360, 360, 720, 120, 120, 180, 30, 1, 5040, 5040, 5040, 5040, 2520, 5040, 2520, 2520, 840, 2520, 840, 210, 420, 42, 1, 40320, 40320, 40320, 40320, 20160, 20160, 40320, 40320, 20160

Offset: 1

Wolfdieter Lang, Jul 13 2007

The order of this array is according to the Abramowitz-Stegun (A-St) ordering of partitions (see A036036).
The row lengths sequence is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
These numbers are similar to M_0, M_1, M_2, M_3, M_4 given in A111786, A036038, A036039, A036040, A117506, respectively.
Combinatorial interpretation: a(n,k) counts the sets of lists (ordered subsets) obtained from partitioning the set {1..n}, with the lengths of the lists given by the k-th partition of n in A-St order. E.g., a(5,5) is computed from the number of sets of lists of lengths [1^1,2^2] (5th partition of 5 in A-St order). Hence a(5,5) = binomial(5,2)*binomial(3,2) = 5!/(1!*2!) = 60 from partitioning the numbers 1,2,...,5 into sets of lists of the type {[.],[..],[..]}.
This array, called M_3(2), is the k=2 member of a family of partition arrays generalizing A036040 which appears as M_3 = M_3(k=1). S2(2) = A105278 (unsigned Lah number triangle) is related to M_3(2) in the same way as S2(1), the Stirling2 number triangle, is related to M_3(1). - Wolfdieter Lang, Oct 19 2007
Another combinatorial interpretation: a(n,k) enumerates unordered forests of increasing binary trees which are described by the k-th partition of n in the Abramowitz-Stegun order. - Wolfdieter Lang, Oct 19 2007
A relation between partition polynomials formed from these "refined Lah numbers" and Lagrange inversion for an o.g.f. is presented in the link "Lagrange a la Lah" along with an e.g.f. and an umbral binary operator tree representation. - Tom Copeland, Apr 12 2011
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133437. - Tom Copeland, Feb 09 2018
Divided by n!, the row partition polynomials are the elementary homogeneous Schur polynomials presented on p. 44 of the Bracci et al. paper. - Tom Copeland, Jun 04 2018
Also presented (renormalized) as the Schur polynomials on p. 19 of the Konopelchenko and Schief paper with associations to differential operators related to the KP hierarchy. - Tom Copeland, Nov 19 2018
Through equation 4.8 on p. 26 of the Arbarello reference, these polynomials appear in the Hirota bilinear equations 4.7 related to tau-function solutions of the KP hierarchy. - Tom Copeland, Jan 21 2019
These partition polynomials appear as Feynman amplitudes in their Bell polynomial guise (put x_n = n!c_n in A036040 for the indeterminates of the Bell polynomials) in Kreimer and Yeats and Balduf (e.g., p. 27). - Tom Copeland, Dec 17 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
These partition polynomials are referred to as Schur functions by Segal and Wilson, who present associations with Plucker coordinates, Grassmannians, and the tau functions of the KdV hierarchy. See pages 51 and 61. - Tom Copeland, Jan 08 2022

			Triangle starts:
  [  1];
  [  2,   1];
  [  6,   6,   1];
  [ 24,  24,  12, 12,  1];
  [120, 120, 120, 60, 60, 20, 1];
  ...
a(5,6) = 20 = 5!/(3!*1!) because the 6th partition of 5 in A-St order is [1^3,2^1].
a(5,5) = 60 enumerates the unordered [1^1,2^2]-forest with 5 vertices (including the three roots) composed of three such increasing binary trees: 5*((binomial(4,2)*2)*(1*2))/2! = 5*12 = 60.

E. Arbarello, "Sketches of KdV", Contemp. Math. 312 (2002), p. 9-69.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
F. Bracci, M. Contreras, S. Díaz-Madrigal, and A. Vasil'ev, Classical and stochastic Löwner-Kufarev equations, arXiv:1309.6423 [math.CV], 2013.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
T. Copeland, Lagrange a la Lah
T. Copeland, Formal group laws and binomial Sheffer sequences, 2018.
T. Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015
T. Ernst, A history of the q-calculus and a new method, 2000.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
Wolfdieter Lang, First 10 rows and more.
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
G. Segal and G. Wilson, Loop groups and equations of KdV type, Publications mathématiques de l’I.H.É.S., tome 61, 1985, p. 5-65.

Cf. A105278 (unsigned Lah triangle |L(n, m)|) obtained by summing the numbers for given part number m.
Cf. A000262 (row sums), identical with row sums of unsigned Lah triangle A105278.
A134133(n, k) = A130561(n, k)/A036040(n, k) (division by the M_3 numbers). - Wolfdieter Lang, Oct 12 2007
Cf. A036039, A263634, A263916.
Cf. A096162.
Cf. A133437.
Cf. A127671.

a(n,k) = n!/(Product_{j=1..n} e(n,k,j)!) with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.
From Tom Copeland, Sep 18 2011: (Start)
Raising and lowering operators are given for the partition polynomials formed from A130561 in the Copeland link in "Lagrange a la Lah Part I" on pp. 22-23.
An e.g.f. for the partition polynomials is on page 3:
  exp[t*:c.*x/(1-c.*x):] = exp[t*(c_1*x + c_2*x^2 + c_3*x^3 + ...)] where :(...): denotes umbral evaluation of the enclosed expression and c. is an umbral coefficient. (End)
From Tom Copeland, Sep 07 2016: (Start)
The row partition polynomials of this array P(n,x_1,x_2,...,x_n), given in the Lang link, are n! * S(n,x_1,x_2,...,x_n), where S(n,x_1,...,x_n) are the elementary Schur polynomials, for which d/d(x_m) S(n,x_1,...,x_n) = S(n-m,x_1,...,x_(n-m)) with S(k,...) = 0 for k < 0, so d/d(x_m) P(n,x_1,...,x_n) = (n!/(n-m)!) P(n-m,x_1,...,x_(n-m)), confirming that the row polynomials form an Appell sequence in the indeterminate x_1 with P(0,...) = 1. See p. 127 of the Ernst paper for more on these Schur polynomials.
With the e.g.f. exp[t * P(.,x_1,x_2,..)] = exp(t*x_1) * exp(x_2 t^2 + x_3 t^3 + ...), the e.g.f. for the partition polynomials that form the umbral compositional inverse sequence U(n,x_1,...,x_n) in the indeterminate x_1 is exp[t * U(.,x_1,x_2,...)] = exp(t*x_1) exp[-(x_2 t^2 + x_3 t^3 + ...)]; therefore, U(n,x_1,x_2,...,x_n) = P(n,x_1,-x_2,.,-x_n), so umbrally P[n,P(.,x_1,-x_2,-x_3,...),x_2,x_3,...,x_n] = (x_1)^n = P[n,P(.,x_1,x_2,...),-x_2,-x_3,...,-x_n]. For example, P(1,x_1) = x_1, P2(x_1,x_2) = 2 x_2 + x_1^2, and P(3,x_1,x_2,x_3) = 6 x_3 + 6 x_2 x_1 + x_1^3, then P[3,P(.,x_1,-x_2,...),x_2,x_3] = 6 x_3 + 6 x_2 P(1,x_1) + P(3,x_1,-x_2,-x_3) = 6 x_3 + 6 x_2 x_1 + 6 (-x_3) + 6 (-x_2) x_1 + x_1^3 = x_1^3.
From the Appell formalism, umbrally [P(.,0,x_2,x_3,...) + y]^n = P(n,y,x_2,x_3,...,x_n).
The indeterminates of the partition polynomials can also be extracted using the Faber polynomials of A263916 with -n * x_n = F(n,S(1,x_1),...,S(n,x_1,...,x_n)) = F(n,P(1,x_1),...,P(n,x_1,...,x_n)/n!). Compare with A263634.
Also P(n,x_1,...,x_n) = ST1(n,x_1,2*x_2,...,n*x_n), where ST1(n,...) are the row partition polynomials of A036039.
(End)

Name augmented by Tom Copeland, Dec 08 2022

A080575 Triangle of multinomial coefficients, read by rows (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1, 1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1, 1, 8, 28, 28, 56, 168, 56, 35, 280, 210, 420, 70, 280, 280, 840, 560, 56, 105, 420, 210, 28, 1, 1, 9, 36, 36, 84, 252, 84, 126, 504, 378, 756, 126, 315, 1260, 1260, 1890, 1260, 126, 280, 2520, 840, 1260, 3780, 1260, 84, 945, 1260, 378, 36, 1, 1, 10, 45, 45, 120, 360, 120, 210, 840, 630, 1260, 210

Offset: 1

Wouter Meeussen, Mar 23 2003

This is different from A036040 and A178867.
T[n,m] = count of set partitions of n with block lengths given by the m-th partition of n in the canonical ordering.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey] and program below)
The complete Bell polynomial of the first n primes gives A007446. (End)
The difference with A036040 and A178867 lies in the ordering of the monomials. This sequence uses lexicographic ordering, while in A036040 the total order (power) of the monomials prevails (Abramowitz-Stegun style): e.g., in row 6 we have ...+ 15*x[3]*x[5] + 15*x[3]*x[6]^2 + 10*x[4]^2 +...; in A036040 the coefficient of x[3]*x[6]^2 would come after that of x[4]^2 because the total order is higher, here it comes before in view of the lexicographic order. - M. F. Hasler, Jul 12 2015

			For n=4 the 5 integer partitions in canonical ordering with corresponding set partitions and counts are:
   [4]       -> #{1234} = 1
   [3,1]     -> #{123/4, 124/3, 134/2, 1/234} = 4
   [2,2]     -> #{12/34, 13/24, 14/23} = 3
   [2,1,1]   -> #{12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34} = 6
   [1,1,1,1] -> #{1/2/3/4} = 1
Thus row 4 is [1, 4, 3, 6, 1].
Triangle begins:
1;
1, 1;
1, 3,  1;
1, 4,  3,  6,  1;
1, 5, 10, 10, 15,  10,  1;
1, 6, 15, 15, 10,  60, 20, 15,  45,  15,  1;
1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1;
...
Row 4 represents 1*k(4)+4*k(3)*k(1)+3*k(2)^2+6*k(2)*k(1)^2+1*k(1)^4 and T(4,4)=6 since there are six ways of partitioning four labeled items into one part with two items and two parts each with one item.

See A036040 for the column labeled "M_3" in Abramowitz and Stegun, Handbook, p. 831.

Alois P. Heinz, Rows n = 1..26, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Berg, Kimmo Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).
R. J. Cano, Sequencer program.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Wikipedia, Cumulant.
Wikipedia, Bell polynomials

See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Row lengths are A000041.
Cf. A007446. - Tilman Neumann, Oct 05 2008
Cf. A178866 and A178867 (version 3). - Johannes W. Meijer, Jun 21 2010
Maximum value in row n gives A102356(n).

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[Apply[Multinomial, IntegerPartitions[w], {1}]/Apply[Times, (runs/@ IntegerPartitions[w])!, {1}], {w, 6}]
(* Second program: *)
completeBellMatrix[x_, n_] := Module[{M, i, j}, M[, ] = 0; For[i=1, i <= n-1 , i++, M[i, i+1] = -1]; For[i=1, i <= n , i++, For[j=1, j <= i, j++, M[i, j] = Binomial[i-1, j-1]*x[i-j+1]]]; Array[M, {n, n}]]; completeBellPoly[x_, n_] := Det[completeBellMatrix[x, n]]; row[n_] := List @@ completeBellPoly[x, n] /. x[] -> 1 // Reverse; Table[row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover, Aug 31 2016, after Tilman Neumann *)
B[0] = 1;
B[n_] := B[n] = Sum[Binomial[n-1, k] B[n-k-1] x[k+1], {k, 0, n-1}]//Expand;
row[n_] := Reverse[List @@ B[n] /. x[_] -> 1];
Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Aug 10 2018, after Wolfdieter Lang *)

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1]:=-1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix(x,n))): end_proc:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

PARI
```
\\ See links.
```

A080575_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))}
A080575_row(n)={(f(s)=if(type(s)!="t_INT",concat(apply(f,select(t->t,Vec(s)))),s))(A080575_poly(n))} \\ M. F. Hasler, Jul 12 2015

A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!).

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -4, -3, 12, -6, 1, -5, -10, 20, 30, -60, 24, 1, -6, -15, -10, 30, 120, 30, -120, -270, 360, -120, 1, -7, -21, -35, 42, 210, 140, 210, -210, -1260, -630, 840, 2520, -2520, 720, 1, -8, -28, -56, -35, 56, 336, 560, 420, 560, -336, -2520, -1680, -5040, -630, 1680, 13440, 10080, -6720

Offset: 1

Wolfdieter Lang, Jan 23 2007

Connected objects from general (disconnected) objects.
The row lengths of this array is p(n):=A000041(n) (partition numbers).
In row n the partitions of n are taken in the Abramowitz-Stegun order.
One could call the unsigned numbers |a(n,k)| M_5 (similar to M_0, M_1, M_2, M_3 and M_4 given in A111786, A036038, A036039, A036040 and A117506, resp.).
The inverse relation (disconnected from connected objects) is found in A036040.
(d/da(1))p_n[a(1),a(2),...,a(n)] = n b_(n-1)[a(1),a(2),...,a(n-1)], where p_n are the partition polynomials of the cumulant generator A127671 and b_n are the partition polynomials for A133314. - Tom Copeland, Oct 13 2012
See notes on relation to Appell sequences in a differently ordered version of this array A263634. - Tom Copeland, Sep 13 2016
Given a binomial Sheffer polynomial sequence defined by the e.g.f. exp[t * f(x)] = Sum_{n >= 0} p_n(t) * x^n/n!, the cumulants formed from these polynomials are the Taylor series coefficients of f(x) multipied by t. An example is the sequence of the Stirling polynomials of the first kind A008275 with f(x) = log(1+x), so the n-th cumulant is (-1)^(n-1) * (n-1)! * t. - Tom Copeland, Jul 25 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
Ignoring signs, these polynomials appear in Schröder in the set of equations (II) on p. 343 and in Stewart's translation on p. 31. - Tom Copeland, Aug 25 2021

			Row n=3: [1,-3,2] stands for the polynomial 1*x[3] - 3*x[1]*x[2] + 2*x[1]^3 (the Abramowitz-Stegun order of the p(3)=3 partitions of n=3 is [3],[1,2],[1^3]).

C. Itzykson and J.-M. Drouffe, Statistical field theory, vol. 2, p. 413, eq.(13), Cambridge University Press, (1989).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Tom Copeland, The creation / raising operators for Appell sequences.
Wolfdieter Lang, First 10 rows of cumulant numbers and polynomials
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen vol. 2, 317-365, 1870.
G. Stewart, On infinitely many algorithms for solving equations, 1993, (translation into English of Schröder's paper above).

Cf. A133314, A263916, A263634.
Cf. A008275.
Cf. A036039, A036040, A130561.

E.g.f. for multivariate row polynomials A(t) := log(1 + Sum_{k>=1} x[k]*(t^k)/k!).
Row n polynomial p_n(x[1],...,x[n]) = [(t^n)/n!]A(t).
a(n,m) = A264753(n, m) * M_3(n,m) with M_3(n,m) = A036040(n,m) (Abramowitz-Stegun M_3 numbers). - corrected by Johannes W. Meijer, Jul 12 2016
p_n(x[1],...,x[n]) = -(n-1)!*F(n,x[1],x[2]/2!,..,x[n]/n!) in terms of the Faber polynomials F(n,b1,..,bn) of A263916. - Tom Copeland, Nov 17 2015
With D = d/dz and M(0) = 1, the differential operator R = z + d(log(M(D))/dD = z + d(log(1 + x[1] D + x[2] D^2/2! + ...))/dD = z + p.*exp(p.D) = z + Sum_{n>=0} p_(n+1)(x[1],..,x[n]) D^n/n! is the raising operator for the Appell sequence A_n(z) = (z + x[.])^n = Sum_{k=0..n} binomial(n,k) x[n-k] z^k with the e.g.f. M(t) e^(zt), i.e., R A_n(z) = A_(n+1)(z) and dA_n(z)/dz = n A_(n-1)(z). The operator Q = z - p.*exp(p.D) generates the Appell sequence with e.g.f. e^(zt) / M(t). - Tom Copeland, Nov 19 2015

A115131 Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, -2, 1, 3, -3, 1, -4, 4, 2, -4, 1, 5, -5, -5, 5, 5, -5, 1, -6, 6, 6, 3, -6, -12, -2, 6, 9, -6, 1, 7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1, -8, 8, 8, 8, 4, -8, -16, -16, -8, -8, 8, 24, 12, 24, 2, -8, -32, -16, 8, 20, -8, 1, 9, -9, -9, -9, -9, 9, 18, 18, 9, 9, 18, 3, -9, -27, -27, -27, -27, -9, 9, 36, 18, 54, 9, -9, -45, -30, 9, 27, -9, 1

Offset: 1

Wolfdieter Lang, Jan 13 2006

N*t^{(N)}n(sigma_1, ..., sigma_N):= sum((x_k)^n, k=1..N) with the elementary symmetric function sigma_k (superscript (N) omitted) in terms of the indeterminates x_1,...,x_N, is an N-variable generalization of Chebyshev's polynomials C_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1, sigma_2 = 1). In general, C_n^{(N)}(sigma_1, ..., sigma{N-1}) := t^{(N)}n(sigma_1, ..., sigma{N-1}, sigma_N:=1). If n > N, one puts sigma_{N+1} = 0, ..., sigma_n = 0.
The sequence of row lengths of this array is A000041(n) (partition numbers).
In row n, this triangular array uses partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in A048996 = |A111786|, A036038, A036039 and A036040, resp.).
Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.
N*t^{(N)}n(sigma_1, ..., sigma_N) gives the sum of the n-th power of the indeterminates x_1, ... , x_N in terms of the elementary symmetric functions of these indeterminates. The coefficient T(n, k) of this partition array corresponds to the k-th partition of n in the Abramowitz-Stegun order, and it multiplies the product of sigma_j functions encoded by this partition. See the example for n = 4 below. - _Wolfdieter Lang, Mar 09 2015

			First few rows of triangle T(n,k) are as follows (see the link for rows 1..10):
   1;
  -2,  1;
   3, -3,  1;
  -4,  4,  2, -4, 1;
   5, -5, -5,  5, 5, -5, 1;
  ...
n=4: N*t^{(N)}_4 = -4*(sigma_4)^1 + 4*(sigma_1)*(sigma_3) + 2*(sigma_2)^2 -4*(sigma_1)^2*(sigma_2) + 1*(sigma_1)^4.
  (For 2 <= N < 4, one puts sigma_{N+1} = 0 = ... = sigma_4 = 0.) This becomes Sum_{k = 1..N} (x_k)^4 if the sigma functions are written in terms of the variables x_1, x_2, ..., x_N. E.g., for N=2: 0 + 0 + 2*(x_1*x_2)^2 -4*(x_1 + x_2)^2*(x_1*x_2) + 1*(x_1 + x_2)^4 = (x_1)^4 + (x_2)^4.

P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 5 (with a_k -> sigma_k).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see pp. 831-832. [alternative scanned copy].
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; see Theorem 1.
Wolfdieter Lang, First 10 rows of the array.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 5 (with a_k -> sigma_k).

Cf. A000041, A000225, A036038, A036039, A036040, A048996, A111786.
Cf. A210258 (in another ordering of partitions), A132460 (N=2), A325477 (N=3),
A324602 (N=4).

T(n,k) = (n/m(n,k))*A111786(n,k) for the k-th partition of n with m(n,k) parts in the Abramowitz-Stegun order for n >= 1 and k = 1..p(n), where p(n) := A000041(n).

Explicitly: T(n,k) = (-1)^(n + m(n,k)) * n * (m(n,k) - 1)!/(Product_{j = 1..n} e(k,j)!), where m(n,k):= Sum_{j = 1..n} e(k,j), with [1^e(k, 1), 2^e(k,2), ..., n^e(k,n)] being the k-th partition of n in the mentioned order. For m(n,k), see A036043.

Various sections edited by Petros Hadjicostas, Dec 14 2019

A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 9, 0, 14, 0, 1, 0, 89, 0, 30, 0, 1, 225, 0, 439, 0, 55, 0, 1, 0, 3429, 0, 1519, 0, 91, 0, 1, 11025, 0, 24940, 0, 4214, 0, 140, 0, 1, 0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1, 893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

The row polynomials t(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy the recurrence relation t(n,x) = x*t(n-1,x) + ((n-1)^2)*t(n-2,x); t(-1,x)=0, t(0,x)=1. - Wolfdieter Lang, see above.
This is an example of a Sheffer triangle (coefficient triangle for Sheffer polynomials). In the umbral calculus (see the Roman reference given under A048854) s(n,x) := Sum_{k=0..n} T(n,k)*x^k would be called Sheffer polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. for column number k>=0 given by (1/sqrt(1-x^2))*((arctanh(x))^k)/k!. The e.g.f. given below is rewritten in this Sheffer context as (1/sqrt(1-x^2))*exp(y*log(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*arctanh(x)). The rows of the Jabotinsky type triangle |A049218| provide the coefficients of the associated polynomials. - Wolfdieter Lang, Feb 24 2005
The solution of the differential-difference relation f(n+1,x)= (d/dx)f(n,x) + (n^2)*f(n-1,x), n >= 1, with inputs f(0,x) and f(1,x) = (d/dx)f(0,x) is f(n,x) = t(n,d_x)*f(0,x), with the differential operator d_x:=d/dx and the row polynomials t(n,x) defined above. This problem appears in a computation of thermo field dynamics where f(0,x)=1/cosh(x). See the triangle A060081. - Wolfdieter Lang, Feb 24 2005
The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - Wolfdieter Lang, Jul 22 2005
T(n,k)=0 if n-k= 1(mod 2), else T(n,k) = sum of M2(n,p), p from {1,...,A000041(n)} restricted to partitions with exactly k odd parts and any nonnegative number of even parts. For the M2-multinomial numbers in A-St order see A036039(n,p). - Wolfdieter Lang, Aug 07 2007

			Triangle begins:
  [1],
  [0, 1],
  [1, 0, 1],
  [0, 5, 0, 1],
  [9, 0, 14, 0, 1],
  [0, 89, 0, 30, 0, 1],
  [225, 0, 439, 0, 55, 0, 1],
  [0, 3429, 0, 1519, 0, 91, 0, 1],
  [11025, 0, 24940, 0, 4214, 0, 140, 0, 1],
  [0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1],
  [893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1],
  [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 385, 0, 1],
  ...
Signed version begins:
  [1],
  [0, 1],
  [-1, 0, 1],
  [0, -5, 0, 1],
  [9, 0, -14, 0, 1],
  [0, 89, 0, -30, 0, 1],
  [-225, 0, 439, 0, -55, 0, 1],
  [0, -3429, 0, 1519, 0, -91, 0, 1],
  ...
From _Peter Bala_, Feb 23 2024: (Start)
Maple can verify the following series for Pi:
Row 1 polynomial R(1, x) = x:
Pi = 3 + 4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(1, 2*n)*R(1, 2*n+2)).
Row 3 polynomial R(3, x) = 5*x + x^3:
(3/2)^2 * Pi = 7 + 4*(3^4)*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(3, 2*n)*R(3, 2*n+2)).
Row 5 polynomial R(5, x) = 89*x + 30*x^3 + x^5:
((3*5)/(2*4))^2 * Pi = 11 + 4*(3*5)^4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(5, 2*n)*R(5, 2*n+2)). (End)

Alois P. Heinz, Rows n = 0..140, flattened
Marcelo Aguiar, Sarah Brauner, and Vic Reiner, Configuration spaces and peak representations, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 13. See p. 12.
Peter Bala, Meixner polynomials and Brouncker's continued fraction for Pi
Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
Adel Hamdi and Jiang Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Meixner polynomial of the first kind.

Cf. A060338, A060523, A094368, A028353 (col 1), A103916 (col 2), A103917 (col 3), A103918 (col 4).

Cf. A111594 (associated Sheffer polynomials), A142979, A142983.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
      `if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015
# alternative
A060524 := proc(n,k)
    option remember;
    if nR. J. Mathar, Jul 06 2023

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
T(n, k) = T(n-1, k-1) + ((n-1)^2)*T(n-2, k); T(-1, k):=0, T(n, -1):=0, T(0, 0)=1, T(n, k)=0 if nWolfdieter Lang, see above.
The Meixner polynomials defined by S_0(x)=1, S_1(x) = x; S_{n+1}(x) = x*S_n(x) - n^2*S_{n-1}(x) give a signed version of this triangle (cf. A060338). - N. J. A. Sloane, May 30 2013
From Peter Bala, Apr 10 2024: (Start)
The n-th row polynomial R(n, x) satisfies
(4*n + 2)*R(n, x) = (x + 1)*R(n, x+2) - (x - 1)*R(n, x-2).
Series for Pi involving the row polynomials R(n, x): for n >= 0 there holds
((2*n + 1)!!/(2^n*n!))^2 * Pi = (4*n + 3) + 4*((2*n + 1)!!^4) * Sum_{k >= 1} (-1)^(k+1)/((2*k + 1)*R(2*n+1, 2*k)*R(2*n+1, 2*k+2)). Cf. A142979 and A142983.
R(2*n, 0) = A001147(n)^2 = A001818(n); R(2*n+1, 0) = 0.
R(n, 1) = n! = A000142(n).
R(2*n, 2) = (4*n + 1)*A001147(n)^2 = (4*n + 1)*((2*n)!/(2^n*n!))^2;
R(2*n+1, 2) = 2*A001447(n+1)^2 = 2*(2*n + 1)!^2/(n!^2*4^n).
R(n, 3) = (2*n + 1)*n! = A007680(n). (End)

cp's OEIS Frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

SageMath

Formula

Extensions

A271703 Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A263916 Coefficients of the Faber partition polynomials.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A130561 Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278; elementary Schur polynomials / functions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs