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A157064
Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.
1
2, 90, 2252, 39500, 535502, 5930022, 55599992, 452715672, 3262336002, 21114177018, 124188986196, 670283877588, 3346707628446, 15564971674518, 67830161708592, 278406848295312, 1081149205136382, 3988232552194662, 14025412751733092, 47171740235162340
OFFSET
1,1
LINKS
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
Sequence in context: A076532 A212301 A226339 * A058527 A342329 A306760
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 22 2009
STATUS
approved