A001725 -id:A001725 - cp's OEIS frontend

A094645 Triangle of generalized Stirling numbers of the first kind.

1, -1, 1, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 2, 1, 0, -6, -5, 5, 5, 1, 0, -24, -26, 15, 25, 9, 1, 0, -120, -154, 49, 140, 70, 14, 1, 0, -720, -1044, 140, 889, 560, 154, 20, 1, 0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1, 0, -40320, -69264, -8540, 50840, 44835, 17913, 3990, 510, 35, 1

Offset: 0

Vladeta Jovovic, May 17 2004

From Wolfdieter Lang, Jun 20 2011: (Start)
The row polynomials s(n,x) := Sum_{j=0..n} T(n,k)*x^k satisfy risefac(x-1,n) = s(n,x), with the rising factorials risefac(x-1,n) := Product_{j=0..n-1} (x-1+j), n >= 1, risefac(x-1,0) = 1. Compare with the formula risefac(x,n) = s1(n,x), with the row polynomials s1(n,x) of A132393 (unsigned Stirling1).
This is the lower triangular Sheffer array with e.g.f.
  T(x,z) = (1-z)*exp(-x*log(1-z)) (the rewritten e.g.f. from the formula section). See the W. Lang link under A006232 for Sheffer matrices and the Roman reference. In the notation which indicates the column e.g.f.s this is Sheffer (1-z,-log(1-z)). In the umbral notation (cf. Roman) this is called Sheffer for (exp(t),1-exp(-t)).
The row polynomials satisfy s(n,x) = (x+n-1)*s(n-1,x), s(0,x)=1, and s(n,x) = (x-1)*s1(n-1,x), n >= 1, s1(0,x) = 1, with the unsigned Stirling1 row polynomials s1(n,x).
The row polynomials also satisfy
  s(n,x) - s(n,x-1) = n*s(n-1,x), n > 1, s(0,x) = 1
  (from the Meixner identity, see the Meixner reference given at A060338).
The row polynomials satisfy as well (from corollary 3.7.2. p. 50 of the Roman reference)
  s(n,x) = (x-1)*s(n-1,x+1), n >= 1, s(0,n) = 1.
The exponential convolution identity is
  s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,y)*s1(n-k,x),
  n >= 0, with symmetry x <-> y.
The row sums are 1 for n=0 and 0 otherwise, and the alternating row sums are 1,-2,2, followed by zeros, with e.g.f. (1-x)^2.
The Sheffer a-sequence Sha(n) = A164555(n)/A027642(n) with e.g.f. x/(1-exp(-x)), and the z-sequence is Shz(n) = -1 with e.g.f. -exp(x).
The inverse Sheffer matrix is ((-1)^(n-k))*A105794(n,k) with e.g.f. exp(z)*exp(x*(1-exp(-z))). (End)
Triangle T(n,k), read by rows, given by (-1, 1, 0, 2, 1, 3, 2, 4, 3, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012
Also coefficients of t in t*(t-1)*Sum[(-1)^(n+m) t^(m-1) StirlingS1[n,m], {m,n}] in which setting t^k equal to k gives n!, from this follows that the dot product of row n with [0,...,n] equals (n-1)!. - Wouter Meeussen, May 15 2012

			Triangle begins
   1;
  -1,   1;
   0,  -1,   1;
   0,  -1,   0,   1;
   0,  -2,  -1,   2,   1;
   0,  -6,  -5,   5,   5,   1;
   0, -24, -26,  15,  25,   9,   1;
   ...
Recurrence:
  -2 = T(4,1) = T(3,0) + (4-2)*T(3,1) = 0 + 2*(-1).
Row polynomials:
  s(3,x) = -x+x^3 = (x-1)*s1(2,x) = (x-1)*(x+x^2).
  s(3,x) = (x-1)*s(2,x+1) = (x-1)*(-(x+1)+(x+1)^2).
  s(3,x) - s(3,x-1) = -x+x^3 -(-(x-1)+(x-1)^3) = 3*(-x+x^2) = 3*s(2,x).

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Cf. A049444, A049458, A094646, A132393, A105794.

A094645_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x-n+2, n)), x, k), k=0..n): seq(print(A094645_row(n)), n=0..6); # Peter Luschny, May 16 2013

t[n_, k_] /; n >= k >= 0 := t[n, k] = t[n-1, k-1] + (n-2)*t[n-1, k]; t[n_, k_] /; n < k = 0; t[, -1] = 0; t[0, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, 10}, {k, 0, n}] ] (* _Jean-François Alcover, Sep 29 2011, after recurrence *);
Table[CoefficientList[t*(t-1)*Sum[(-1)^(n+m)*t^(m-1)*StirlingS1[n,m],{m,n}],t],{n,1,7}] (* Wouter Meeussen, May 15 2012 *)

E.g.f.: (1-y)^(1-x).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively. - Philippe Deléham, Nov 13 2007
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then |T(n,i)| = |f(n,i,-1)|, for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
From Wolfdieter Lang, Jun 20 2011: (Start)
T(n,k) = |S1(n-1,k-1)| - |S1(n-1,k)|, n >= 1, k >= 1, with |S1(n,k)| = A132393(n,k) (unsigned Stirling1).
Recurrence: T(n,k) = T(n-1,k-1) + (n-2)*T(n-1,k) if n >= k >= 0; T(n,k) = 0 if n < k; T(n,-1) = 0; T(0,0) = 1.
E.g.f. column k: (1-x)*((-log(1-x))^k)/k!. (End)
T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i, m = 1 for n >= 0. See A130534, A370518 for m=0 and m=2. - Igor Victorovich Statsenko, Feb 27 2024

A051338 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -6, 1, 42, -13, 1, -336, 146, -21, 1, 3024, -1650, 335, -30, 1, -30240, 19524, -5000, 635, -40, 1, 332640, -245004, 74524, -11985, 1075, -51, 1, -3991680, 3272688, -1139292, 218344, -24885, 1687, -63, 1, 51891840, -46536624, 18083484, -3977764, 541849, -46816, 2506, -76, 1

Offset: 0

Wolfdieter Lang

a(n,m)= ^6P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(6+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(6*t),exp(t)-1).

			{1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).

Cf. A000035 A084938.

a051338 n k = a051338_tabl !! n !! k
a051338_row n = a051338_tabl !! n
a051338_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 6)
-- Reinhard Zumkeller, Mar 11 2014

t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*Pochhammer[6, k]*StirlingS1[n - k, i], {k, 0, n - i}]; Flatten[Table[t[n, i], {n, 0, 8}, {i, 0, n}]][[1 ;; 45]] (* Jean-François Alcover, Jun 01 2011, after Milan Janjic *)

a(n, m)= a(n-1, m-1) - (n+5)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^6).

Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,6), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

A051339 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -7, 1, 56, -15, 1, -504, 191, -24, 1, 5040, -2414, 431, -34, 1, -55440, 31594, -7155, 805, -45, 1, 665280, -434568, 117454, -16815, 1345, -57, 1, -8648640, 6314664, -1961470, 336049, -34300, 2086, -70, 1, 121080960, -97053936, 33775244, -6666156, 816249, -63504, 3066, -84, 1

Offset: 0

Wolfdieter Lang

a(n,m)= ^7P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(7+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(7*t),exp(t)-1).

			{1}; {-7,1}; {56,-15,1}; {-504,191,-24,1}; ... s(2,x)= 56-15*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

The first (m=0) column sequence is A001730. Row sums (signed triangle): A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n).

Cf. A000035 A084938.

a051339 n k = a051339_tabl !! n !! k
a051339_row n = a051339_tabl !! n
a051339_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 7)
-- Reinhard Zumkeller, Mar 11 2014

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^7, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

a(n, m)= a(n-1, m-1) - (n+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^7).

Triangle (signed) = [ -7, -1, -8, -2, -9, -3, -10, -4, -11, -5, ...] DELTA A000035; triangle (unsigned) = [7, 1, 8, 2, 9, 3, 10, 4, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,7), for n=1,2,...;i=0...n. [From Milan Janjic, Dec 21 2008]

A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 42, 1, 144, 168, 336, 1, 600, 2940, 1680, 3024, 1, 4950, 33600, 35280, 18144, 30240, 1, 26586, 336630, 717360, 444528, 211680, 332640, 1, 234528, 4870992, 11313120, 10329984, 5927040, 2661120, 3991680

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 = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144356.
Same partition product with length statistic is A049374.
Diagonal a(A000217(n)) = rising_factorial(6,n-1), A001725(n+4).
Row sum is A049402.

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

Cf. A157385, A157384, A157383, A157400, A126074, 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-4).

A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(x) = x + 3.

Original entry on oeis.org

1, 3, 3, 9, 24, 9, 27, 141, 141, 27, 81, 726, 1410, 726, 81, 243, 3471, 11406, 11406, 3471, 243, 729, 15828, 81327, 136872, 81327, 15828, 729, 2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187, 6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561, 19683, 1277619, 19489380, 105311556, 237144642, 237144642, 105311556, 19489380, 1277619, 19683

Offset: 0

Dale Gerdemann, Apr 17 2015

			Array t(n,k) begins as:
    1,     3,       9,        27,         81,         243, ... A000244;
    3,    24,     141,       726,       3471,       15828, ...;
    9,   141,    1410,     11406,      81327,      533259, ...;
   27,   726,   11406,    136872,    1390521,    12609198, ...;
   81,  3471,   81327,   1390521,   19467294,   237144642, ...;
  243, 15828,  533259,  12609198,  237144642,  3794314272, ...;
  729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     24,       9;
    27,    141,     141,       27;
    81,    726,    1410,      726,       81;
   243,   3471,   11406,    11406,     3471,      243;
   729,  15828,   81327,   136872,    81327,    15828,     729;
  2187,  69873,  533259,  1390521,  1390521,   533259,   69873,   2187;
  6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000244, A008292, A001725 (row sums), A038221, A256890, A257606.
Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.
Similar sequences listed in A256890.

f[n_]:= n+3;
t[n_, k_]:= t[n,k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1,k] +f[n]*t[n,k-1]]];
T[n_, k_]= t[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

f(x) = x + 3;
T(n, k) = t(n-k, k);
t(n, m) = {if (!n && !m, return(1)); if (n < 0 || m < 0, return (0)); f(m)*t(n-1,m) + f(n)*t(n,m-1);}
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ Michel Marcus, Apr 23 2015

def f(n): return n+3
@CachedFunction
def t(n,k):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
def A257627(n,k): return t(n-k,k)
flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022

T(n,k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
Sum_{k=0..n} T(n, k) = A001725(n+5).
From G. C. Greubel, Feb 22 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A049374 A triangle of numbers related to triangle A030527.

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984

Offset: 1

Wolfdieter Lang

a(n,1) = A001725(n+4). a(n,m)=: S1p(6; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m) = A008275 (unsigned Stirling first kind), S1p(2; n,m) = A008297(n,m) (unsigned Lah numbers). S1p(3; n,m) = A046089(n,m), S1p(4; n,m) = A049352, S1p(5; n,m) = A049353(n,m).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} (a(n,m)*x^m), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j >= 1 come in j+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007

			Triangle begins
       1;
       6,       1;
      42,      18,      1;
     336,     276,     36,     1;
    3024,    4200,    960,    60,    1;
   30240,   66024,  23400,  2460,   90,   1;
  332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Muniru A Asiru, Table of n, a(n) for n = 1..1275
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A049402 (row sums), A134140 (alternating row sums).

Flat(List([1..10],n->Factorial(n)*List([1..n],k->Sum([1..k],j->(-1)^(k-j)*Binomial(k,j)*Binomial(n+5*j-1,5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+5)!/120, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
(* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+5)!/120&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k)=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1),j,1,k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n,k)=(n!*sum(j=1,k,(-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1)))/(5^k*k!);
for(n=1,12,for(k=1,n,print1(a(n,k),", "));print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */

a(n, m) = n!*A030527(n, m)/(m!*5^(n-m)); a(n, m) = (5*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n < m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!.

a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 8, 3, 1, 60, 40, 15, 4, 1, 360, 240, 90, 24, 5, 1, 2520, 1680, 630, 168, 35, 6, 1, 20160, 13440, 5040, 1344, 280, 48, 7, 1, 181440, 120960, 45360, 12096, 2520, 432, 63, 8, 1, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 9, 1

Offset: 1

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Row sums are the factorial numbers (A000142). First column is A001710.
T(n,k) = number of permutations of [n] in which 1,2,...,k is a subsequence but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] in which 12 is a subsequence but 123 is not. - Emeric Deutsch, Nov 12 2004
T(n,k) is the number of deco polyominoes of height n with k cells in the last column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column). - Emeric Deutsch, Jan 06 2005
T(n,k) is the number of permutations p of [n] for which the smallest i such that p(i)Emeric Deutsch, Feb 23 2008
Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). - Emeric Deutsch, Feb 23 2008
Matrix inverse of A128227*A154990. - Mats Granvik, Feb 08 2009
Differences in the columns of A173333 which counts the n-permutations with an initial ascending run of length at least k. - Geoffrey Critzer, Jun 18 2017
The triangle with each row reversed is A130477. - Michael Somos, Jun 25 2017

			T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
     1;
     1,    1;
     3,    2,   1;
    12,    8,   3,   1;
    60,   40,  15,   4,  1;
   360,  240,  90,  24,  5,  1;
  2520, 1680, 630, 168, 35,  6,  1;
  ...

M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.

Alois P. Heinz, Rows n = 1..141, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Colin Defant and James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
Emeric Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

Cf. A002260. - Gary W. Adamson, Feb 22 2012
Cf. A000166, A002627, A055596, A092580, A092956, A130477.
Cf. A000027, A001710, A001715, A001720, A001725, A005563.
Cf. A000522.

Flat(List([1..11],n->Concatenation([1],List([1..n-1],k->Factorial(n)*k/Factorial(k+1))))); # Muniru A Asiru, Jun 10 2018

A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
[A092582(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 06 2022

Drop[Drop[Abs[Map[Select[#, # < 0 &] &, Map[Differences, nn = 10; Range[0, nn]! CoefficientList[Series[(Exp[y x] - 1)/(1 - x), {x, 0, nn}], {x, y}]]]], 1], -1] // Grid (* Geoffrey Critzer, Jun 18 2017 *)

{T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* Michael Somos, Jun 25 2017 */

def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Sep 06 2022

T(n, k) = n!*k/(k+1)! for k

Inverse of:

-1, 1;

-1, -2, 1;

-1, -2, -3, 1;

-1, -2, -3, -4, 1;

... where A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Feb 22 2012

T(2n,n) = A092956(n-1) for n>0. - Alois P. Heinz, Jun 19 2017

From Alois P. Heinz, Dec 17 2021: (Start)

Sum_{k=1..n} k * T(n,k) = A002627(n).

|Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)

From G. C. Greubel, Sep 06 2022: (Start)

T(n, 1) = A001710(n).

T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.

T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.

T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.

T(n, n-1) = A000027(n-1).

T(n, n-2) = A005563(n-1), n >= 3. (End)

Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - Alois P. Heinz, Apr 28 2023

A049460 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -5, 1, 30, -11, 1, -210, 107, -18, 1, 1680, -1066, 251, -26, 1, -15120, 11274, -3325, 485, -35, 1, 151200, -127860, 44524, -8175, 835, -45, 1, -1663200, 1557660, -617624, 134449, -17360, 1330, -56, 1, 19958400, -20355120, 8969148, -2231012, 342769, -33320, 2002, -68, 1

Offset: 0

Wolfdieter Lang

a(n,m)= ^5P_n^m in the notation of the given reference with a(0,0) := 1.
The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(5+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(5*t),exp(t)-1).

			{1}; {-5,1}; {30,-11,1}; {-210,107,-18,1}; ... s(2,x)= 30-11*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5).

Cf. A000035 A084938.

a049460 n k = a049460_tabl !! n !! k
a049460_row n = a049460_tabl !! n
a049460_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 5)
-- Reinhard Zumkeller, Mar 11 2014

a[n_, m_] := Pochhammer[m+1, n-m] SeriesCoefficient[Log[1+x]^m/(1+x)^5, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

a(n, m)= a(n-1, m-1) - (n+4)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,5), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Second formula corrected by Philippe Deléham, Nov 10 2008

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A111532 Row 5 of table A111528.

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A094645 Triangle of generalized Stirling numbers of the first kind.

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Maple

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A111533 Row 6 of table A111528.

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A051338 Generalized Stirling number triangle of first kind.

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A051339 Generalized Stirling number triangle of first kind.

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A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

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A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(x) = x + 3.