A144381 a(n) = A142458(n+5, n). Fifth diagonal of A142458 triangle.

1, 677, 47175, 1709675, 44451550, 947113254, 17716715490, 302925749370, 4856552119935, 74258231957275, 1095758678253041, 15736592058221517, 221321453958111620, 3062416225698505060, 41836761536767296660, 565817483249269872324, 7591501608353930033805, 101209790951020335444705

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..890

Index entries for linear recurrences with constant coefficients, signature (75, -2520, 50204, -661458, 6086718, -40273648, 194372208, -687083013, 1771618303, -3293261472, 4325310828, -3886563008, 2261691264, -765434880, 114150400).

Formula

a(n) = A142458(n+5, n).

From G. C. Greubel, Mar 16 2022: (Start)

G.f.: x*(1 +602*x -1080*x^2 -172614*x^3 +1780275*x^4 -5025348*x^5 -7549548*x^6 +60043488*x^7 -99645984*x^8 +39979520*x^9 +27596800*x^10)/((1-x)^5*(1-4*x)^4*(1-7*x)^3*(1-10*x)^2*(1-13*x)).

a(n) = (1/1944)*((27*n^3 +216*n^2 +549*n +440)*(3*n +2 - 2*4^(n+4)) +

60*(9*n^2 +57*n +88)*7^(n+3) -32*(3*n+11)*10^(n+4) + 880*13^(n+3)). (End)

Mathematica

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A144381[n_]:= T[n+4, n, 3];
Table[A144381[n], {n, 30}] (* modified by G. C. Greubel, Mar 16 2022 *)

Prog

(Magma) [(1/1944)*((27*n^3 + 216*n^2 + 549*n + 440)*(3*n + 2 - 2*4^(n + 4)) +
   60*(9*n^2 + 57*n + 88)*7^(n + 3) - 32*(3*n + 11)*10^(n + 4) +
880*13^(n + 3)): n in [1..30]]; // G. C. Greubel, Mar 16 2022
(Sage)
@CachedFunction
def T(n, k, m): # A144381
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A144381(n): return T(n+4, n, 3)
[A144381(n) for n in (1..30)] # G. C. Greubel, Mar 16 2022

Crossrefs

Cf. A142458, A142976, A144380, A144414.

Sequence in context: A205749 A108824 A205471 * A097773 A248887 A031524

Adjacent sequences: A144378 A144379 A144380 * A144382 A144383 A144384

Keyword

nonn

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Extensions

Edited by G. C. Greubel, Mar 16 2022

Status

approved