A157061 Number of integer sequences of length n+1 with sum zero and sum of absolute values 24.

2, 72, 1442, 20260, 220250, 1958460, 14768810, 96900810, 563873400, 2953859370, 14097919968, 61908797418, 252208679268, 959882556570, 3433533723900, 11603837100660, 37221177046410, 113779617228060, 332648955112250, 933146517188760, 2518877938240202

Offset: 1

Author

R. H. Hardin, Feb 22 2009

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (25,-300,2300,-12650,53130,-177100,480700,-1081575, 2042975,-3268760,4457400,-5200300,5200300,-4457400,3268760,-2042975,1081575, -480700,177100,-53130,12650,-2300,300,-25,1).

Crossrefs

Cf. A103881, A156554.

Programs

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
A157061[n_]:= A103881[n, 12];
Table[A157061[n], {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

Sage

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157061(n): return A103881(n, 12)
[A157061(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

Formula

a(n) = T(n,12); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).

From G. C. Greubel, Jan 24 2022: (Start)

a(n) = (n+1)*binomial(n+11, 12)*Hypergeometric3F2([-11, -n, 1-n], [2, -n-11], 1).

a(n) = (2704156/24!)*n*(n+1)*(19120211066880000 + 40213832085504000*n + 61866024285081600*n^2 + 47770238895160320*n^3 + 35477403021764352*n^4 + 15129353226246336*n^5 + 7138320279252096*n^6 + 1926081009812080*n^7 + 648411230685152*n^8 + 117787792143956*n^9 + 30215435337736*n^10 + 3799367698665*n^11 + 775177128207*n^12 + 67808650591*n^13 + 11342892341*n^14 + 678888650*n^15 + 95251222*n^16 + 3725106*n^17 + 446226*n^18 + 10285*n^19 + 1067*n^20 + 11*n^21 + n^22).

G.f.: 2*x*(1 + 11*x + 121*x^2 + 605*x^3 + 3025*x^4 + 9075*x^5 + 27225*x^6 + 54450*x^7 + 108900*x^8 + 152460*x^9 + 213444*x^10 + 213444*x^11 + 213444*x^12 + 152460*x^13 + 108900*x^14 + 54450*x^15 + 27225*x^16 + 9075*x^17 + 3025*x^18 + 605*x^19 + 121*x^20 + 11*x^21 + x^22)/(1-x)^25. (End)