A157066 Number of integer sequences of length n+1 with sum zero and sum of absolute values 34.

2, 102, 2892, 57460, 882030, 11053434, 117206264, 1078467624, 8774904690, 64062783510, 424600608564, 2579499722124, 14479567043214, 75613799423610, 369504358622640, 1698353774374704, 7375213677918294, 30379740299612514, 119122913376492980, 446056011713374860

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 (35,-595,6545,-52360,324632,-1623160,6724520, -23535820,70607460,-183579396,417225900,-834451800,1476337800,-2319959400, 3247943160,-4059928950,4537567650,-4537567650,4059928950,-3247943160,2319959400, -1476337800,834451800,-417225900,183579396,-70607460,23535820,-6724520,1623160, -324632,52360,-6545,595,-35,1).

Formula

a(n) = T(n,17); 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+16, 17)*Hypergeometric3F2([-16, -n, 1-n], [2, -n-16], 1).

a(n) = (2333606220/34!)*n*(n+1)*(7441973323855715893248000000 + 18155084795637437929881600000*n + 30268626521952180908851200000*n^2 + 27504128369891325149577216000*n^3 + 22380511931408981359868313600*n^4 + 11606451235232148856801198080*n^5 + 6053325843616709826370609152*n^6 + 2071495721724703057714876416*n^7 + 776772176331488107582976256*n^8 + 188575401978015909077544960*n^9 + 54249004662342491124700928*n^10 + 9739700938346246478267904*n^11 + 2242198636428402181902944*n^12 + 305221374822225945324800*n^13 + 57932851765719841948880*n^14 + 6064778909442097812240*n^15 + 970512936702416581665*n^16 + 78610569988240809600*n^17 + 10791805239981923160*n^18 + 675564468731071680*n^19 + 80680394732550780*n^20 + 3869168748681600*n^21 + 406620563860680*n^22 + 14666674470240*n^23 + 1369455578790*n^24 + 35960795520*n^25 + 3007754088*n^26 + 54285504*n^27 + 4095964*n^28 + 45440*n^29 + 3112*n^30 + 16*n^31 + n^32).

G.f.: 2*x*(1 + 16*x + 256*x^2 + 1920*x^3 + 14400*x^4 + 67200*x^5 + 313600*x^6 + 1019200*x^7 + 3312400*x^8 + 7949760*x^9 + 19079424*x^10 + 34978944*x^11 + 64128064*x^12 + 91611520*x^13 + 130873600*x^14 + 147232800*x^15 + 165636900*x^16 + 147232800*x^17 + 130873600*x^18 + 91611520*x^19 + 64128064*x^20 + 34978944*x^21 + 19079424*x^22 + 7949760*x^23 + 3312400*x^24 + 1019200*x^25 + 313600*x^26 + 67200*x^27 + 14400*x^28 + 1920*x^29 + 256*x^30 + 16*x^31 + x^32)/(1-x)^35. (End)

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
A157066[n_]:= A103881[n, 17];
Table[A157066[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 A157066(n): return A103881(n, 17)
[A157066(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

Crossrefs

Cf. A103881, A156554.

Sequence in context: A173640 A111012 A256516 * A202740 A283624 A055693

Adjacent sequences: A157063 A157064 A157065 * A157067 A157068 A157069

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved