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

1,1

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A103881, A156554.

Sequence in context: A030993 A283568 A070775 * A179957 A221549 A289197

Adjacent sequences: A157058 A157059 A157060 * A157062 A157063 A157064

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved