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

Author

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

Keywords

• nonn

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).

Crossrefs

Cf. A142458, A142976, A144380, A144414.

Programs

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

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 *)

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

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 [phone]*x^7 -99645984*x^8 [phone]*x^9 [phone]*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)

Extensions

Edited by G. C. Greubel, Mar 16 2022