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A080248 Stirling-like number triangle defined by sequence A000217. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 15 08:18 EDT 2021. Contains 342977 sequences. (Running on oeis4.)