A157065 Number of integer sequences of length n+1 with sum zero and sum of absolute values 32.

2, 96, 2562, 47920, 692610, 8174544, 81659522, 708113304, 5431848930, 37403270520, 233931828834, 1341750437352, 7114703302434, 35117045235720, 162298598439330, 705951252118284, 2903050518427962, 11331495633292524, 42132555868774010, 149703679118108220

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 (33,-528,5456,-40920,237336,-1107568,4272048, -13884156,38567100,-92561040,193536720,-354817320,573166440,-818809200, 1037158320,-1166803110,1166803110,-1037158320,818809200,-573166440,354817320, -193536720,92561040,-38567100,13884156,-4272048,1107568,-237336,40920,-5456, 528,-33,1).

Formula

a(n) = T(n,16); 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 25 2022: (Start)

a(n) = (n+1)*binomial(n+15, 16)*Hypergeometric3F2([-15, -n, 1-n], [2, -n-15], 1).

a(n) = (601080390/32!)*n*(n+1)*(27360196043587190784000000 + 65137211981397216460800000*n + 107110050449356033228800000*n^2 + 94817527804050105212928000*n^3 + 75961411427539608595660800*n^4 + 38202458280851158526730240*n^5 + 19587950887554046039781376*n^6 + 6463560689425876180435200*n^7 + 2379792991631228553219840*n^8 + 553304095999692103772160*n^9 + 156114125142340061791744*n^10 + 26624540206135314300000*n^11 + 6005394587432947709600*n^12 + 768878902291539639600*n^13 + 142854837644598236640*n^14 + 13893755540913698625*n^15 + 2174500936993696575*n^16 + 161097628663020825*n^17 + 21612664028370855*n^18 + 1212359721607125*n^19 + 141388292047275*n^20 + 5907926749725*n^21 + 605873224515*n^22 + 18281995875*n^23 + 1664663325*n^24 + 34287435*n^25 + 2794869*n^26 + 35175*n^27 + 2585*n^28 + 15*n^29 + n^30).

G.f.: 2*x*(1 + 15*x + 225*x^2 + 1575*x^3 + 11025*x^4 + 47775*x^5 + 207025*x^6 + 621075*x^7 + 1863225*x^8 + 4099095*x^9 + 9018009*x^10 + 15030015*x^11 + 25050025*x^12 + 32207175*x^13 + 41409225*x^14 + 41409225*x^15 + 41409225*x^16 + 32207175*x^17 + 25050025*x^18 + 15030015*x^19 + 9018009*x^20 + 4099095*x^21 + 1863225*x^22 + 621075*x^23 + 207025*x^24 + 47775*x^25 + 11025*x^26 + 1575*x^27 + 225*x^28 + 15*x^29 + x^30)/(1-x)^33. (End)

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
A157065[n_]:= A103881[n, 16];
Table[A157065[n], {n, 50}] (* G. C. Greubel, Jan 25 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 A157065(n): return A103881(n, 16)
[A157065(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

Crossrefs

Cf. A103881, A156554.

Sequence in context: A282436 A069121 A362406 * A123115 A119696 A264542

Adjacent sequences: A157062 A157063 A157064 * A157066 A157067 A157068

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved