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

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 (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).

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

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)