A054655 -id:A054655 - cp's OEIS frontend

A177938 Triangle T(n,k) = (-1)^(k+n)*A054655(n,n-k), 0<=k

1, 1, 1, 6, 5, 1, 60, 47, 12, 1, 840, 638, 179, 22, 1, 15120, 11274, 3325, 485, 35, 1, 332640, 245004, 74524, 11985, 1075, 51, 1, 8648640, 6314664, 1961470, 336049, 34300, 2086, 70, 1, 259459200, 188204400, 59354028, 10630508, 1182769, 83720

Offset: 0

Roger L. Bagula, May 15 2010

			[0]         1;
[1]         1,         1;
[2]         6,         5,        1;
[3]        60,        47,       12,        1;
[4]       840,       638,      179,       22,       1;
[5]     15120,     11274,     3325,      485,      35,     1;
[6]    332640,    245004,    74524,    11985,    1075,    51,    1;
[7]   8648640,   6314664,  1961470,   336049,   34300,  2086,   70,  1;
[8] 259459200, 188204400, 59354028, 10630508, 1182769, 83720, 3682, 92, 1;

Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

An unsigned, row-reversed variant of A054655.
Row sums are apparently A001813.
Cf. A001813, A054655.

p[x_, n_] = If[n == 0, 1, (n - 1)! / FunctionExpand[Beta[x + n, n]]];
Table[CoefficientList[p[x, n], x], {n, 0, 8}] // Flatten (* rewritten by Peter Luschny, Mar 22 2022 *)

Row generating function: Gamma(x+2*n)/Gamma(x+n) = Sum_{k>=0} T(n,k)*x^k.

T(n, k) = n!*(-1)^k*[x^k] hypergeom([-n, -x + n - 1], [-n], 1). - Peter Luschny, Mar 22 2022

A008276 Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -6, 11, -6, 1, -10, 35, -50, 24, 1, -15, 85, -225, 274, -120, 1, -21, 175, -735, 1624, -1764, 720, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 1, -36, 546, -4536, 22449, -67284, 118124, -109584, 40320, 1, -45

Offset: 1

N. J. A. Sloane

n-th row of the triangle = charpoly of an (n-1) X (n-1) matrix with (1,2,3,...) in the diagonal and the rest zeros. - Gary W. Adamson, Mar 19 2009
From Daniel Forgues, Jan 16 2016: (Start)
For n >= 1, the row sums [of either signed or absolute values] are
  Sum_{k=1..n} T(n,k) = 0^(n-1),
  Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)
The moment generating function of the probability density function p(x, m=q, n=1, mu=q) = q^q*x^(q-1)*E(x, q, 1)/(q-1)!, with q >= 1, is M(a, m=q, n=1, mu=q) = Sum_{k=0..q}(A000312(q) / A000142(q-1)) * A008276(q, k) * polylog(k, a) / a^q , see A163931 and A274181. - Johannes W. Meijer, Jun 17 2016
Triangle of coefficients of the polynomial x(x-1)(x-2)...(x-n+1), also denoted as falling factorial (x)n, expanded into decreasing powers of x. - _Ralf Stephan, Dec 11 2016

			3!*binomial(x,3) = x*(x-1)*(x-2) = x^3 - 3*x^2 + 2*x.
Triangle begins
  1;
  1,  -1;
  1,  -3,   2;
  1,  -6,  11,   -6;
  1, -10,  35,  -50,  24;
  1, -15,  85, -225, 274, -120;
...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), p. 257.

T. D. Noe, Rows n=0..100 of triangle, 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].
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See pp. 10, 12.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Wikipedia, Stirling numbers and exponential generating functions

See A008275 and A048994, which are the main entries for this triangle of numbers.
See A008277 triangle of Stirling numbers of the second kind, S2(n,k).
Cf. A054654, A054655, A084938, A145324, A094216, A003422, A000166, A000110, A000204, A000045, A000108.

a008276 n k = a008276_tabl !! (n-1) !! (k-1)
a008276_row n = a008276_tabl !! (n-1)
a008276_tabl = map init $ tail a054654_tabl
-- Reinhard Zumkeller, Mar 18 2014

seq(seq(coeff(expand(n!*binomial(x,n)),x,j),j=n..1,-1),n=1..15); # Robert Israel, Jan 24 2016
A008276 := proc(n, k): combinat[stirling1](n, n-k+1) end: seq(seq(A008276(n, k), k=1..n), n=1..9); # Johannes W. Meijer, Jun 17 2016

len = 47; m = Ceiling[Sqrt[2*len]]; t[n_, k_] = StirlingS1[n, n-k+1]; Flatten[Table[t[n, k], {n, 1, m}, {k, 1, n}]][[1 ;; len]] (* Jean-François Alcover, May 31 2011 *)
Flatten@Table[CoefficientList[Product[1-k x, {k, 1, n}], x], {n, 0, 8}] (* Oliver Seipel, Jun 14 2024 *)
Flatten@Table[Coefficient[Product[x-k, {k, 0, n-1}], x, Reverse@Range[n]], {n, Range[9]}] (* Oliver Seipel, Jun 14 2024, after  Ralf Stephan *)

T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),n-k+1))

T(n,k)=if(n<1,0,n!*polcoeff(polcoeff(y*(1+y*x+x*O(x^n))^(1/y),n),k))

def T(n,k): return falling_factorial(x,n).expand().coefficient(x,n-k+1) # Ralf Stephan, Dec 11 2016

n!*binomial(x, n) = Sum_{k=1..n-1} T(n, k)*x^(n-k).
|A008276(n, k)| = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]; A008276(n, k) = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...]. Here DELTA is the operator defined in A084938. - Philippe Deléham, Dec 30 2003
|T(n, k)| = Sum_{m=0..n} A008517(k, m+1)*binomial(n+m, 2*(k-1)), n >= k >= 1. A008517 is the second-order Eulerian triangle. See the Graham et al. reference p. 257, eq. (6.44).
A094638 formula for unsigned T(n, k).
|T(n, k)| = Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*binomial(n-1, k-1+m) if n >= k >= 1, else 0. - Wolfdieter Lang, Sep 12 2005, see A112486.
|T(n, k)| = (f(n-1, k-1)/(2*(k-1))!)* Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*f(2*(k-1), k-1-m)*f(n-k, m) if n >= k >= 1, else 0, where f(n, k) stands for the falling factorial n*(n-1)*...*(n-(k-1)) and f(n, 0):=1. - Wolfdieter Lang, Sep 12 2005, see A112486.
With P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = (1-t)*(1-2*t)*...*(1-(n-1)t) and P(0,t) = 1, exp(P(.,t)*x) = (1+t*x)^(1/t) . Compare A094638. T(n,k+1) = (1/k!) (D_t)^k (D_x)^n ( (1+t*x)^(1/t) - 1 ) evaluated at t=x=0 . - Tom Copeland, Dec 09 2007
Product_{i=1..n} (x-i) = Sum_{k=0..n} T(n,k)*x^k. - Reinhard Zumkeller, Dec 29 2007
E.g.f.: Sum_{n>=0} (Sum_{k=0..n} T(n,n-k)*t^k)/n!) = Sum_{n>=0} (x)n * t^k/n! = exp(x * log(1+t)), with (x)_n the n-th falling factorial polynomial. - _Ralf Stephan, Dec 11 2016
Sum_{j=0..m} T(m, m-j)*s2(j+k+1, m) =  m^k, where s2(j, m) are Stirling numbers of the second kind. - Tony Foster III, Jul 25 2019
For n>=2, Sum_{k=1..n} k*T(n,k) = (-1)^(n-1)*(n-2)!. - Zizheng Fang, Dec 27 2020

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0

Offset: 0

N. J. A. Sloane, Apr 18 2000

Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - Philippe Deléham, Dec 30 2006
From Doudou Kisabaka, Dec 18 2009: (Start)
The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)

			Matrix begins:
  1, 0,  0,  0,  0,   0,    0,     0,      0, ...
  0, 1, -1,  2, -6,  24, -120,   720,  -5040, ...
  0, 0,  1, -3, 11, -50,  274, -1764,  13068, ...
  0, 0,  0,  1, -6,  35, -225,  1624, -13132, ...
  0, 0,  0,  0,  1, -10,   85,  -735,   6769, ...
  0, 0,  0,  0,  0,   1,  -15,   175,  -1960, ...
  0, 0,  0,  0,  0,   0,    1,   -21,    322, ...
  0, 0,  0,  0,  0,   0,    0,     1,    -28, ...
  0, 0,  0,  0,  0,   0,    0,     0,      1, ...
  ...
Triangle begins:
  1;
  1,   0;
  1,  -1,   0;
  1,  -3,   2,    0;
  1,  -6,  11,   -6,    0;
  1, -10,  35,  -50,   24,     0;
  1, -15,  85, -225,  274,  -120,   0;
  1, -21, 175, -735, 1624, -1764, 720, 0;
  ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Eric Weisstein's World of Mathematics, Rising Factorial.
Eric Weisstein's World of Mathematics, FallingFactorial.

Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.

a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
-- Reinhard Zumkeller, Mar 18 2014

a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)
Table[StirlingS1[n,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)

PARI
```
T(n,k)=polcoeff(n!*binomial(x,n), n-k)
```

n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

Additional comments from Thomas Wieder, Dec 29 2006

Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008

A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 5, 6, 1, -2, 11, 14, 24, 1, -5, 25, 5, 94, 120, 1, -9, 55, -75, 304, 444, 720, 1, -14, 112, -350, 1099, 364, 3828, 5040, 1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320, 1, -27, 366, -2646, 12873, -31563, 79064, 34236, 270576, 362880

Offset: 0

N. J. A. Sloane, Apr 17 2000

Apparently A190782 with reversed rows. - Mathew Englander, May 17 2014

			The first few polynomials are:
  1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
So the triangle begins:
  1;
  1,   1;
  1,   1,   2;
  1,   0,   5,     6;
  1,  -2,  11,    14,   24;
  1,  -5,  25,     5,   94,   120;
  1,  -9,  55,   -75,  304,   444,   720;
  1, -14, 112,  -350, 1099,   364,  3828,  5040;
  1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

T(2*n,n) gives A347987.

Cf. A054649, A054655, A054654, A190782 .

c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 9}] // Flatten  (* Jean-François Alcover, Oct 04 2012 *)

T(n, k) = Sum_{i=0..k} Stirling1(i+n-k,n-k)*n!/(i+n-k)!. - Igor Victorovich Statsenko, May 27 2024

Missing 0 corrected by Steve Marak - N. J. A. Sloane, Jul 27 2012

A054649 Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -3, 4, 1, -9, 32, -36, 1, -18, 131, -426, 528, 1, -30, 375, -2370, 7544, -9600, 1, -45, 865, -8955, 52414, -163800, 213120, 1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240, 1, -84, 3122, -66696, 893249, -7664916, 41096908, -125747664, 167973120

Offset: 0

N. J. A. Sloane, Apr 16 2000

			Triangle begins:
  1;
  1,   0;
  1,  -3,    4;
  1,  -9,   32,    -36;
  1, -18,  131,   -426,    528;
  1, -30,  375,  -2370,   7544,    -9600;
  1, -45,  865,  -8955,  52414,  -163800,  213120;
  1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240;
  ...
From _Peter Luschny_, Nov 27 2021: (Start)
The row reversed triangle can be seen as the coefficients of a sequence of monic polynomials with monomials sorted in ascending order which start:
[0]     1;
[1]              x;
[2]     4 -    3*x +      x^2;
[3]   -36 +   32*x -    9*x^2 +     x^3;
[4]   528 -  426*x +  131*x^2 -  18*x^3 +    x^4;
[5] -9600 + 7544*x - 2370*x^2 + 375*x^3 - 30*x^4 + x^5; (End)

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A008275, A008276, A048994, A054651, A054655.

# Some older Maple versions are known to have a bug in the hypergeom function.
with(ListTools): with(PolynomialTools):
CoeffList := p -> op(Reverse(CoefficientList(simplify(p), x))):
p := k -> k!*hypergeom([-k, -x + k], [-k], -1):
seq(CoeffList(p(k)), k = 0..8); # Peter Luschny, Nov 27 2021

c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x-n, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 8}] // Flatten  (* Jean-François Alcover, Oct 04 2012 *)

row(n) = Vec(n!*sum(k=0, n, binomial(x-n, k))); \\ Seiichi Manyama, Sep 24 2021

T(n, k) = n! * [x^(n - k)] hypergeom([-n, -x + n], [-n], -1). - Peter Luschny, Nov 27 2021

cp's OEIS Frontend

A177938 Triangle T(n,k) = (-1)^(k+n)*A054655(n,n-k), 0<=k

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A008276 Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Sage

Formula

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A054649 Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula