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A157061 Number of integer sequences of length n+1 with sum zero and sum of absolute values 24. 1
2, 72, 1442, 20260, 220250, 1958460, 14768810, 96900810, 563873400, 2953859370, 14097919968, 61908797418, 252208679268, 959882556570, 3433533723900, 11603837100660, 37221177046410, 113779617228060, 332648955112250, 933146517188760, 2518877938240202 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 09:20 EDT 2022. Contains 356993 sequences. (Running on oeis4.)