A157064 Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.

2, 90, 2252, 39500, 535502, 5930022, 55599992, 452715672, 3262336002, 21114177018, 124188986196, 670283877588, 3346707628446, 15564971674518, 67830161708592, 278406848295312, 1081149205136382, 3988232552194662, 14025412751733092, 47171740235162340

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 (31,-465,4495,-31465,169911,-736281,2629575, -7888725,20160075,-44352165,84672315,-141120525,206253075,-265182525,300540195, -300540195,265182525,-206253075,141120525,-84672315,44352165,-20160075,7888725, -2629575,736281,-169911,31465,-4495,465,-31,1).

Formula

a(n) = T(n,15); 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+14, 15)*Hypergeometric3F2([-14, -n, 1-n], [2, -n-14], 1).

a(n) = (155117520/30!)*n*(n+1)*(114000816848279961600000 + 264279998869470904320000*n + 428198206877484244992000*n^2 + 368310644587032673075200*n^3 + 290167678780290006589440*n^4 + 141041429579778368449536*n^5 + 71004668064572092241664*n^6 + 22493711118572061653376*n^7 + 8120370606956264477184*n^8 + 1797910570397283902560*n^9 + 496779939204280228640*n^10 + 79886837991962961960*n^11 + 17626771834821917040*n^12 + 2101988853205045350*n^13 + 381651017327064975*n^14 + 34037459504198850*n^15 + 5201044031664375*n^16 + 346174867450230*n^17 + 45303425489595*n^18 + 2220034746930*n^19 + 252351294195*n^20 + 8844405570*n^21 + 883381005*n^22 + 20963670*n^23 + 1857765*n^24 + 26754*n^25 + 2121*n^26 + 14*n^27 + n^28).

G.f.: 2*x*(1 + 14*x + 196*x^2 + 1274*x^3 + 8281*x^4 + 33124*x^5 + 132496*x^6 + 364364*x^7 + 1002001*x^8 + 2004002*x^9 + 4008004*x^10 + 6012006*x^11 + 9018009*x^12 + 10306296*x^13 + 11778624*x^14 + 10306296*x^15 + 9018009*x^16 + 6012006*x^17 + 4008004*x^18 + 2004002 x^19 + 1002001*x^20 + 364364*x^21 + 132496*x^22 + 33124*x^23 + 8281*x^24 + 1274*x^25 + 196*x^26 + 14*x^27 + x^28)/(1-x)^31. (End)

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
A157064[n_]:= A103881[n, 15];
Table[A157064[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 A157064(n): return A103881(n, 15)
[A157064(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

Crossrefs

Cf. A103881, A156554.

Sequence in context: A076532 A212301 A226339 * A058527 A342329 A306760

Adjacent sequences: A157061 A157062 A157063 * A157065 A157066 A157067

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved