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
Examples
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.
Links
- Vincenzo Librandi, Rows n = 0..50, flattened
Programs
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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
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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 *)
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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 */
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
Comments