A080248 Stirling-like number triangle defined by sequence A000217.

1, 1, 1, 1, 4, 1, 1, 13, 10, 1, 1, 40, 73, 20, 1, 1, 121, 478, 273, 35, 1, 1, 364, 2989, 3208, 798, 56, 1, 1, 1093, 18298, 35069, 15178, 1974, 84, 1, 1, 3280, 110881, 368988, 262739, 56632, 4326, 120, 1, 1, 9841, 668566, 3800761, 4310073, 1452011, 177760

Offset

0,5

Comments

Columns include A003462, A016211, A021514. The defining sequence A000217(n) = C(n+1,2) is the sequence of partial sums of the sequence (0,1,2,3,4,...) which defines the Stirling numbers of the second kind A008277.

n-th row = M^n * [1,0,0,0,...], where M = an infinite lower triangular matrix with (1, 3, 6, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal. - Gary W. Adamson, Apr 13 2009

Row sums = A124373 starting (1, 2, 6, 25, 135, ...). - Gary W. Adamson, Jul 11 2011

Links

Vincenzo Librandi, Rows n = 0..50, flattened

Formula

Columns are generated by 1/Product_{k=1..n+1} (1 - C(k + 1, 2)*x). [In other words:

T(n, k) = [x^(n-k)] 1/Product_{j=0..k+2}(1 - x*binomial(j, 2)).]

T(n, k) = (k*(k+1)/2) * T(n-1,k) + T(n-1,k-1), T(n,n)=1. - Vladimir Kruchinin, Aug 25 2020

T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * (2*i + 3) * binomial(2*k + 3,k-i) * ((i+1) * (i+2) / 2)^(n+1)) * 2^(k+1) / (2*k + 3)! for 0 <= k <= n. - Werner Schulte, Oct 29 2020

The polynomials p(n,x) = Sum_{k=0..n} T(n,k) * (k!*(k+1)!/2^k) * x^(k+2) satisfy for n >= 0 the equations p(n+1,x) = p(1,x) * p''(n,x) / 2 and p(n,-1) = 0^n when p'' is the second derivative of p. - Werner Schulte, Dec 15 2020

Example

Rows are
  {1},
  {1,  1},
  {1,  4,  1},
  {1, 13, 10,  1},
  {1, 40, 73, 20, 1},
  ...
For example, 73 = 13 + 6*10, 20 = 10 + 10*1.

Maple

gf  := k -> 1/mul(1 - x*j*(j-1)/2, j=0..k+2):
ser := k -> series(gf(k), x, 16):
T := (n, k) -> coeff(ser(k), x, n-k):
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 29 2020

Mathematica

max = 10; t[n_, n_] = n*(n+1)/2; t[n_, k_] /; k == n-1 = 1; t[_, _] = 0; m = Table[t[n, k], {n, 1, max}, {k, 1, max}]; row[n_] := MatrixPower[m, n][[All, 1]]; Table[Take[row[n], n+1], {n, 0, max-1}] // Flatten (* Jean-François Alcover, Jun 25 2013, after Gary W. Adamson *)

Prog

(PARI) {T(n, k) = local(s); if( k<0 || k>n, 0, forvec(v = vector(n-k, i, [0, k]), s += prod(i=1, n-k, v[i] * (v[i] + 1) / 2), 1)); s}; /* Michael Somos, Feb 06 2004 */

Crossrefs

Cf. A000217, A008277, A124373.

Sequence in context: A039755 A247502 A047874 * A139382 A157180 A179086

Adjacent sequences: A080245 A080246 A080247 * A080249 A080250 A080251

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 17 2003

Status

approved