A049403 -id:A049403 - cp's OEIS frontend

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is also the number of n X n symmetric permutation matrices.
a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017
Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008
a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson, Apr 22 2003
Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - Roberto E. Martinez II, Jan 09 2002
The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... (A001147 with interpolated zeros). - Philippe Deléham, Jun 10 2005
Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006
a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan, Mar 07 2006
Equals row sums of triangle A152736 starting with offset 1. - Gary W. Adamson, Dec 12 2008
Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - Emeric Deutsch, Jun 08 2009
Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - Joerg Arndt, Jun 28 2009
Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - Joerg Arndt, Jul 29 2012
Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - Joerg Arndt, Nov 11 2012
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - David Callan, Aug 27 2014
Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015
a(n) is the number of elements x in the symmetric group S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 [Edited on Jul 24 2025]
a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020
From Thomas Anton, Apr 20 2020: (Start)
Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
From Iosif Pinelis, Mar 12 2021: (Start)
a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)

			Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - _Henry Bottomley_, Jan 16 2001
G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(4) = 10 standard Young tableaux:
  1 2 3 4
.
  1 2   1 3   1 2 3   1 2 4   1 3 4
  3 4   2 4   4       3       2
.
  1 2   1 3   1 4
  3     2     2
  4     4     3
.
  1
  2
  3
  4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
             {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                        {{1,3},{2}}    {{1,4},{2,3}}
                        {{1},{2},{3}}  {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{4}}
(End)

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Wikipedia, Telephone numbers.
Wikipedia, Young tableau.
Index entries for "core" sequences
Index entries for sequences related to Young tableaux.
Index entries for related partition-counting sequences

Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
See also A005425 for another version of the switchboard problem.
Equals 2 * A001475(n-1) for n>1.
First column of array A099020.
A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
Cf. A152736, A128229. - Gary W. Adamson, Dec 12 2008
Diagonal of A182172. - Alois P. Heinz, May 30 2012
Cf. A001813, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018
Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.

a000085 n = a000085_list !! n
  a000085_list = 1 : 1 : zipWith (+)
    (zipWith (*) [1..] a000085_list) (tail a000085_list) -- Reinhard Zumkeller, May 16 2013

A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # Zerinvary Lajos, Sep 24 2007
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # Zerinvary Lajos, Jun 11 2008

<Roger L. Bagula, Oct 06 2006 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *)
Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *)
a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 18 2017, after Emanuele Munarini *)
RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* Joan Ludevid, Jun 17 2022 *)
sds[{}]:={{}};sds[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sds[Complement[set,s]]]/@Cases[Subsets[set,{1,2}],{i,_}]; Table[Length[sds[Range[n]]],{n,0,10}] (* Gus Wiseman, Jan 11 2021 *)

B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
  a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
  makelist(a(n),n,0,40); /* Emanuele Munarini, May 16 2014 */

makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */

N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013

from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<>1)+1)) # Chai Wah Wu, Aug 31 2023

A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
[simplify(A000085(n)) for n in range(28)] # Peter Luschny, Aug 21 2014

def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
for n in range(100): print(n, a85(n)) # Manfred Scheucher, Jan 07 2018

D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004
For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - Max Alekseyev, Oct 31 2015
The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
For asymptotics see the Robinson paper.
a(n) = Sum_{m=0..n} A099174(n,m). - Roger L. Bagula, Oct 06 2006
O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Gary W. Adamson, Dec 29 2008: (Start)
a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End)
a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011
Row sums of |A096713|. a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A047974 and A080599. - Peter Bala, Dec 07 2011
From Sergei N. Gladkovskii, Dec 03 2011 - Oct 28 2013: (Start)
Continued fractions:
E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
a(n) ~ (1/sqrt(2)) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 16 2014
a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014
0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Aug 22 2014
From Peter Bala, Oct 06 2021: (Start)
a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)

A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 3, 4, 1, 0, 0, 1, 6, 5, 1, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 7, 56, 126, 120, 55, 12, 1

Offset: 1

Wolfdieter Lang

A convolution triangle of numbers obtained from A019590.
a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).
The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if nRiordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - Paul Barry, Feb 02 2005 [with offset 0]
Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212 and 1122. - Emeric Deutsch, Apr 05 2005 [see MacMahon and Riordan. - Wolfdieter Lang, Jul 27 2023]
Subtriangle of A026729. - Philippe Deléham, Aug 31 2006
a(n,k) is the number of length n-1 binary sequences having no two consecutive 0's with exactly k-1 1's. Example: a(6,4)=6 because we have 01011, 01101, 01110, 10101, 10110, 11010. - Geoffrey Critzer, Jul 22 2013
Mirrored, shifted Fibonacci polynomials of A011973. The polynomials (illustrated below) of this entry have the property that p(n,t) = t * [p(n-1,t) + p(n-2,t)]. The additive properties of Pascal's triangle (A007318) are reflected in those of these polynomials, as can be seen in the Example Section below and also when the o.g.f. G(x,t) below is expanded as the series x*(1+x) + t * [x*(1+x)]^2 + t^2 * [x*(1+x)]^3 + ... . See also A053122 for a relation to Cartan matrices. - Tom Copeland, Nov 04 2014
Rows of this entry appear as columns of an array for an infinitesimal generator presented in the Copeland link. - Tom Copeland, Dec 23 2015
For n >= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - Eric W. Weisstein, Apr 10 2017
With an additional initial matrix element a_(0,0) = 1 and column of zeros a_(n,0) = 0 for n > 0, these are antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Cf. A011973. And A169803. - Tom Copeland, Jul 02 2018

			Triangle starts:
  [ 1]  1
  [ 2]  1   1
  [ 3]  0   2   1
  [ 4]  0   1   3   1
  [ 5]  0   0   3   4   1
  [ 6]  0   0   1   6   5   1
  [ 7]  0   0   0   4  10   6   1
  [ 8]  0   0   0   1  10  15   7   1
  [ 9]  0   0   0   0   5  20  21   8   1
  [10]  0   0   0   0   1  15  35  28   9   1
  [11]  0   0   0   0   0   6  35  56  36  10   1
  [12]  0   0   0   0   0   1  21  70  84  45  11   1
  [13]  0   0   0   0   0   0   7  56 126 120  55  12   1
  ...
From _Tom Copeland_, Nov 04 2014: (Start)
For quick comparison to other polynomials:
  p(1,t) = 1
  p(2,t) = 1 + 1 t
  p(3,t) = 0 + 2 t + 1 t^2
  p(4,t) = 0 + 1 t + 3 t^2 + 1 t^3
  p(5,t) = 0 + 0   + 3 t^2 + 4 t^3 +  1 t^4
  p(6,t) = 0 + 0   + 1 t^2 + 6 t^3 +  5 t^4 +  1 t^5
  p(7,t) = 0 + 0   + 0     + 4 t^3 + 10 t^4 +  6 t^5 + 1 t^6
  p(8,t) = 0 + 0   + 0     + 1 t^3 + 10 t^4 + 15 t^5 + 7 t^6 + 1 t^7
  ...
Reading along columns gives rows for Pascal's triangle. (End)

P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one), Chelsea Publishing Company, New York, 1960, Vol. I, nr. 124, p. 151.
John Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, London, 1958. eq. (35), p.124, 11. p. 154.

Indranil Ghosh, Rows 1..125 of triangle, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 18.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Addendum to Elliptic Lie Triad
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 21.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139.
Peter J. Olver, The canonical contact form.
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.

Cf. A104597, A146559, A155020, A125145, A057078, A039717, A001792, A057862, A011973, A115139.

/* As triangle */ [[Binomial(k, n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Nov 05 2014

for n from 1 to 12 do seq(binomial(k,n-k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 05 2005

nn=10;CoefficientList[Series[(1+x)/(1-y x - y x^2),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Jul 22 2013 *)
Table[Binomial[k, n - k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)
CoefficientList[Table[x^(n/2 - 1) Fibonacci[n + 1, Sqrt[x]], {n, 10}],
   x] // Flatten (* Eric W. Weisstein, Apr 10 2017 *)

a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: (x*(1+x))^m.

As a number triangle with offset 0, this is T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2k+1). The antidiagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 21 2004

G.f.: (1 + x)/(1 - y*x - y*x^2). - Geoffrey Critzer, Jul 22 2013 [offset 0] [with offset 1: g.f. of row polynomials in y: x*(1+x)*y/(1 - x*(1+x)*y). - Wolfdieter Lang, Jul 27 2023]

From Tom Copeland, Nov 04 2014: (Start)

O.g.f: G(x,t) = x*(1+x) / [1 - t*x*(1+x)] = -P[Cinv(-x),t], where P(x,t)= x / (1 + t*x) and Cinv(x)= x*(1-x) are the compositional inverses in x of Pinv(x,t) = -P(-x,t) = x / (1 - t*x) and C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108.

Therefore, Ginv(x,t) = -C[Pinv(-x,t)] = {-1 + sqrt[1 + 4*x/(1+t*x)]}/2, which is -A124644(-x,t).

This places this array in a family of arrays related by composition of P and C and their inverses and interpolation by t, such as A091867 and A104597, and associated to the Catalan, Motzkin, Fine, and Fibonacci numbers. Cf. A104597 (polynomials shifted in t) A125145, A146559, A057078, A000045, A155020, A125145, A039717, A001792, A057862, A011973, A115139. (End)

More terms from Emeric Deutsch, Apr 05 2005

A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55, 0, 1

Offset: 0

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004

Absolute values of A066325.
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006
Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008
Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013
The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014
Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014
From James East, Aug 17 2015: (Start)
T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016
Antidiagonals of this entry are rows of A001497. - Tom Copeland, Oct 04 2016
This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018
See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021

			h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
From _Paul Barry_, Nov 06 2008: (Start)
Triangle begins
   1,
   0,  1,
   1,  0,  1,
   0,  3,  0,  1,
   3,  0,  6,  0,  1,
   0, 15,  0, 10,  0,  1,
  15,  0, 45,  0, 15,  0,  1
Production array starts
  0, 1,
  1, 0, 1,
  0, 2, 0, 1,
  0, 0, 3, 0, 1,
  0, 0, 0, 4, 0, 1,
  0, 0, 0, 0, 5, 0, 1 (End)

Alois P. Heinz, Rows n = 0..150, flattened
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Paul Barry, Riordan array, orthogonal polynomials as moments, and Hankel transforms, arXiv:1102.0921 [math.CO], 2011.
G.-S. Cheon, J.-H. Jung and L. W. Shapiro, Generalized Bessel numbers and some combinatorial settings, Discrete Math., 313 (2013), 2127-2138.
T. Copeland, Juggling Zeros in the Matrix (Example II), 2020.
James East and Robert D. Gray, Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, arXiv:1404.2359 [math.GR], 2014. See Theorem 8.4 and Table 7. - _James East_, Aug 17 2015
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 2. - _N. J. A. Sloane_, May 29 2014
S. Licciardi, Umbral Calculus, a Different Mathematical Language, arXiv:1803.03108 [math.CA], 2018.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 329 and A137286. From _Tom Copeland_, Jan 03 2016).
R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011
S. Yang and Z. Qiao, The Bessel numbers and Bessel matrices, Jrn. Math. Rsch. and Exposition, July 2011, Vol. 31, No. 4, pp.627-636. DOI:10.3770/j.issn:1000-341X.2011.04.006.

Row sums (polynomial values at x=1) are A000085.
Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4).
Cf. A000217, A001147, A059343, A060821, A066325.
Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Cf. A137286.
Cf. A001497.
Cf. A111062, A344678.

T:=proc(n,k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006

nn=10;a=y x+x^2/2!;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, May 08 2012 *)
H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)

T(n,k)=if(k<=n && k==Mod(n,2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014

import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

def Trow(n: int) -> list[int]:
    row: list[int] = [0] * (n + 1); row[n] = 1
    for k in range(n - 2, -1, -2):
        row[k] = (row[k + 2] * (k + 2) * (k + 1)) // (n - k)
    return row  # Peter Luschny, Jan 08 2023

def A099174_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)*M[n-1,k+1]
    return M
A099174_triangle(9)  # Peter Luschny, Oct 06 2012

h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).
T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012
T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014
The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014
From Tom Copeland, Dec 13 2015: (Start)
The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839.
Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0).
As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x).
For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016
h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018
From Tom Copeland, Jun 06 2021: (Start)
In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2.
e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials.
The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)

A122848 Exponential Riordan array (1, x(1+x/2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0

Offset: 0

Paul Barry, Sep 14 2006

Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013.
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins:
    1
    0    1
    0    1    1
    0    0    3    1
    0    0    3    6    1
    0    0    0   15   10    1
    0    0    0   15   45   15    1
    0    0    0    0  105  105   21    1
    0    0    0    0  105  420  210   28    1
    0    0    0    0    0  945 1260  378   36    1
From _Gus Wiseman_, Jan 12 2021: (Start)
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
  {{1},{2,3},{4,5}}  {{1},{2},{3},{4,5}}  {{1},{2},{3},{4},{5}}
  {{1,2},{3},{4,5}}  {{1},{2},{3,4},{5}}
  {{1,2},{3,4},{5}}  {{1},{2,3},{4},{5}}
  {{1,2},{3,5},{4}}  {{1,2},{3},{4},{5}}
  {{1},{2,4},{3,5}}  {{1},{2},{3,5},{4}}
  {{1},{2,5},{3,4}}  {{1},{2,4},{3},{5}}
  {{1,3},{2},{4,5}}  {{1},{2,5},{3},{4}}
  {{1,3},{2,4},{5}}  {{1,3},{2},{4},{5}}
  {{1,3},{2,5},{4}}  {{1,4},{2},{3},{5}}
  {{1,4},{2},{3,5}}  {{1,5},{2},{3},{4}}
  {{1,4},{2,3},{5}}
  {{1,4},{2,5},{3}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}
In the second case, we have:
  {{1,2},{3,4}}  {{1,2}}  {}
  {{1,2},{3,5}}  {{1,3}}
  {{1,2},{4,5}}  {{1,4}}
  {{1,3},{2,4}}  {{1,5}}
  {{1,3},{2,5}}  {{2,3}}
  {{1,3},{4,5}}  {{2,4}}
  {{1,4},{2,3}}  {{2,5}}
  {{1,4},{2,5}}  {{3,4}}
  {{1,4},{3,5}}  {{3,5}}
  {{1,5},{2,3}}  {{4,5}}
  {{1,5},{2,4}}
  {{1,5},{3,4}}
  {{2,3},{4,5}}
  {{2,4},{3,5}}
  {{2,5},{3,4}}
(End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
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.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
S. Yang and Z. Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From _Tom Copeland_, Dec 18 2013]

Cf. A001497, A008275, A039755, A096713, A104556, A113278, A130757.
Row sums are A000085.
Column sums are A001515.
Same as A049403 but with a first column k = 0.
The same set partitions counted by number of pairs are A100861.
Reversing rows gives A111924 (without column k = 0).
A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A320732, A321729, A339741, A339887.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 27 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Second program: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 23 2018,after Peter Luschny *)
sbs[{}]:={{}};sbs[set:{i_,_}]:=Join@@Function[s,(Prepend[#1,s]&)/@sbs[Complement[set,s]]]/@Cases[Subsets[set],{i}|{i,_}];
Table[Length[Select[sbs[Range[n]],Length[#]==k&]],{n,0,6},{k,0,n}] (* Gus Wiseman, Jan 12 2021 *)

{T(n,k)=if(2*kn, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */

# uses[inverse_bell_transform from A265605]
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
From Daniel Checa, Aug 28 2022: (Start)
E.g.f. for the m-th column: (x^2/2+x)^m/m!.
T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)

A096713 Irregular triangle T(n,k) of nonzero coefficients of the modified Hermite polynomials (n >= 0 and 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 3, -6, 1, 15, -10, 1, -15, 45, -15, 1, -105, 105, -21, 1, 105, -420, 210, -28, 1, 945, -1260, 378, -36, 1, -945, 4725, -3150, 630, -45, 1, -10395, 17325, -6930, 990, -55, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, 135135, -270270, 135135, -25740, 2145, -78, 1

Offset: 0

Eric W. Weisstein, Jul 04 2004

Triangle of nonzero coefficients of matching polynomial of complete graph of order n.
Row sums of absolute values produce A000085 (number of involutions). - Wouter Meeussen, Mar 12 2008
Row n has floor(n/2) + 1 nonzero coefficients. - Robert Israel, Dec 23 2015
Also the nonzero terms of the Bell matrix generated by the sequence [-1,1,0,0,0, ...] read by rows (see second Sage program). For the definition of the Bell matrix see A264428. - Peter Luschny, Jan 20 2016
From Petros Hadjicostas, Oct 28 2019: (Start)
The formulas about the p.d.f. of the standard normal distribution were proved, for example, by Charlier (1905, pp. 13-15), but they were well-known for many years before him. Charlier (1905) has generalized these results to other measures whose n-th moment (around 0) exists for each integer n >= 0.
Different forms (with or without signs) of these coefficients T(n,k) appear in other arrays as well; e.g., see A049403, A104556, A122848, A130757 (odd rows only), etc.
(End)

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
    1;
    1;
   -1,     1;
   -3,     1;
    3,    -6,    1;
   15,    -10,   1;
  -15,     45, -15,   1;
  -105,   105, -21,   1;
   105,  -420, 210, -28, 1;
   945, -1260, 378, -36, 1;
   ...
The corresponding modified Hermite polynomials are as follows
He_0(x) = 1, He_1(x) = x,
He_2(x) = -1 + x^2, He_3(x) = -3*x + x^3,
He_4(x) = 3 - 6*x^2 + x^4, He_5(x) = 15*x - 10*x^3 + x^5, ...
[Modified by _Petros Hadjicostas_, Oct 28 2019]

C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened)
Carl V. L. Charlier, Über die Darstellung willkürlicher Funktionen, Arkiv För Matematik, Astronomi Och Fysik, Band 2, No. 20 (Meddelande från Lunds Astronomiska Observatorium, Series I, No. 27), 1905, 1-35. [Accessible only in the USA via the HathiTrust Digital Library.]
Nicolas Loizeau, Berislav Buča, and Dries Sels, Opening Krylov space to access all-time dynamics via dynamical symmetries, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Eric Weisstein's World of Mathematics, Hermite Polynomial.
Eric Weisstein's World of Mathematics, Matching Polynomial. - _Eric W. Weisstein_, Sep 27 2008

Cf. A000085, A049403, A104556, A122848, A130757, A264428.

A:= NULL:
for n from 0 to 20 do
  HH:= expand(orthopoly[H](n,x/sqrt(2))/2^(n/2));
  C:= subs(0=NULL, [seq(coeff(HH,x,j),j=0..n)]);
  A:= A, op(C);
od:
A; #  Robert Israel, Dec 23 2015
# Alternatively:
A096713 := (n, k) -> `if`(2*kA096713(n, k), k=0..n), n=0..13); # Peter Luschny, Dec 24 2015

Table[CoefficientList[HermiteH[n,x/Sqrt[2] ]/2^(n/2),x],{n,0,25}] (* Wouter Meeussen, Mar 12 2008 *)

T(n,k)=if(k<0||2*k>n, 0, (-1)^(n\2-k)*n!/(n\2-k)!/(n%2+2*k)!/2^(n\2-k)) /* Michael Somos, Jun 04 2005 */

from sympy import hermite, Poly, sqrt
def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

from sage.functions.hypergeometric import closed_form
def A096713_row(n):
    R. = ZZ[]
    h = hypergeometric([-n/2,(1-n)/2], [], -2*z)
    T = R(closed_form(h)).coefficients()
    return T[::-1]
for n in range(13): A096713_row(n) # Peter Luschny, Aug 21 2014

# uses[bell_transform from A264428]
def bell_zero_filter(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]
    return [i for f in F for i in f]
print(bell_zero_filter(lambda n: [1,-1][n] if n < 2 else 0, 14)) # Peter Luschny, Jan 20 2016

G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - Wouter Meeussen, Mar 12 2008
From Robert Israel, Dec 23 2015: (Start)
T(2*m, k) = (-1)^(m+k)*(2*m)!*2^(k-m)/((m-k)!*(2*k)!), k = 0..m.
T(2*m+1, k) = (-1)^(m+k)*(2*m+1)!*2^(k-m)/((m-k)!*(2*k+1)!), k = 0..m. (End)
From Petros Hadjicostas, Oct 28 2019: (Start)
Let He_n(x) be the n-th modified Hermite polynomial (see the references above); i.e., let He_n(x) = Sum_{k = 0..m} T(2*m, k)*x^(2*k) when n = 2*m and He_n(x) = Sum_{k = 0..m} T(2*m+1, k)*x^(2*k+1) when n = 2*m+1.
Let phi(x) = (1/sqrt(2*Pi)) * exp(-x^2/2) be the p.d.f. of a standard normal distribution. Then He_n(x) = (-1)^n * (1/phi(x)) * d^n(phi(x))/dx^n for n >= 0.
We have He_n(x) = x*He_{n-1}(x) - (n-1)*He_{n-2}(x) for n >= 2. (End)

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

Original entry on oeis.org

1, 3, -1, 15, -10, 1, 105, -105, 21, -1, 945, -1260, 378, -36, 1, 10395, -17325, 6930, -990, 55, -1, 135135, -270270, 135135, -25740, 2145, -78, 1, 2027025, -4729725, 2837835, -675675, 75075, -4095, 105, -1, 34459425, -91891800, 64324260, -18378360, 2552550, -185640, 7140, -136

Offset: 0

Wolfdieter Lang, Jul 13 2007

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.
The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
(End)
Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

			[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, 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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
T. Copeland, Juggling Zeros in the Matrix: Example II, 2020.
Wolfdieter Lang, First ten rows and more.

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).
Row sums (signed) give A131441. Row sums (unsigned) give A066224.
Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.

seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2,n-m)/m!,m=0..n),n=0..10); # Robert Israel, Dec 25 2015

Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)

a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015
From Tom Copeland, Dec 14 2015: (Start)
Production matrix is
   3,   -1;
  -6,    7,   -1;
   0,  -20,   11,   -1;
   0,    0,  -42,   15,   -1;
   0,    0,    0,  -72,   19,   -1;
   0,    0,    0,    0, -110,   23,   -1;
   0,    0,    0,    0,    0, -156,   27,   -1;
   0,    0,    0,    0,    0,    0, -210,   31,   -1;
   0,    0,    0,    0,    0,    0,    0, -272,   35,   -1;
  ... (End)

Title formula corrected by Tom Copeland, Dec 12 2015

A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 3, 0, 1, 9, 0, 0, 1, 25, 0, 0, 0, 1, 75, 0, 0, 0, 0, 1, 231, 0, 0, 0, 0, 0, 1, 763, 0, 0, 0, 0, 0, 0, 1, 2619, 0, 0, 0, 0, 0, 0, 0, 1, 9495, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35695, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 140151

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144357.
Same partition product with length statistic is A049403.
Diagonal a(A000217(n)) = falling_factorial(1,n-1), row in A008279.
Row sum is A000085.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+3).

A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1

Offset: 0

Paul D. Hanna, Mar 14 2005

Exponential Riordan array [1 - x, x - x^2/2]; cf. A049403. - Peter Bala, Apr 08 2013

Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Rows begin:
   1;
  -1,  1;
   0, -3,   1;
   0,  3,  -6,    1;
   0,  0,  15,  -10,    1;
   0,  0, -15,   45,  -15,     1;
   0,  0,   0, -105,  105,   -21,     1;
   0,  0,   0,  105, -420,   210,   -28,   1;
   0,  0,   0,    0,  945, -1260,   378, -36,   1;
   0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497:
    1;
    1,   1;
    3,   3,   1;
   15,  15,   6,   1;
  105, 105,  45,  10,  1;
  945, 945, 420, 105, 15, 1; ...

G. C. Greubel, Rows n=0..35 of triangle, flattened
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]

Row sums are found in A001464 (offset 1).
Absolute row sums equal A000085.
Cf. A001497, A049403, A096713, A122848, A130757.

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016

E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425

Offset: 0

Paul Barry, Apr 12 2010

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

Showing 1-10 of 10 results.

cp's OEIS Frontend

A144357 Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;n,m) (generalized Stirling triangle).

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A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

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A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

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Magma

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A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

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A122848 Exponential Riordan array (1, x(1+x/2)).

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A096713 Irregular triangle T(n,k) of nonzero coefficients of the modified Hermite polynomials (n >= 0 and 0 <= k <= floor(n/2)).

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A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.