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 (43,-903,12341,-123410,962598,-6096454,32224114, -145008513,563921995,-1917334783,5752004349,-15338678264,36576848168, -78378960360,151532656696,-265182149218,421171648758,-608359048206, 800472431850,-960566918220,1052049481860,-1052049481860,960566918220, -800472431850,608359048206,-421171648758,265182149218,-151532656696, 78378960360,-36576848168,15338678264,-5752004349,1917334783,-563921995, 145008513,-32224114,6096454,-962598,123410,-12341,903,-43,1).
FORMULA
a(n) = T(n,21); 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+20, 21)*Hypergeometric3F2([-20, -n, 1-n], [2, -n-20], 1).
a(n) = (538257874440/42!)*n*(n+1)*(124299255809188481393766275481600000000 + 328816118350366025460284915712000000000*n + 574832876343430323089683765002240000000*n^2 + 568701882574952901291417659454259200000*n^3 + 488065218731065719417147635733626880000*n^4 + 279248916577588134058859235459858432000*n^5 + 154338522148314741971664420691673088000*n^6 + 59227959344696504761998117194266705920*n^7 + 23633263646950664110615399338389323776*n^8 + 6557497087812561104289290673945292800*n^9 + 2014840321470361119845933104915307520*n^10 + 422701102488339328367203562820695040*n^11 + 104284338041749995423701069631220992*n^12 + 17025052804207868558201481522726400*n^13 + 3473748992461285895698788698610560*n^14 + 449827918639409055961252979192960*n^15 + 77602697715487702683123150572128*n^16 + 8071528554520601160114398770800*n^17 + 1197769342263854188918636742220*n^18 + 100831028153769404548233777380*n^19 + 13049306298068383096447853569*n^20 + 892237320110273631864787000*n^21 + 101851737197591285675901050*n^22 + 5654771034611195278152900*n^23 + 574799001272234774582445*n^24 + 25804389773082709176000*n^25 + 2354558801452942771200*n^26 + 84727960701572097480*n^27 + 6988357410140155794*n^28 + 198659321097901200*n^29 + 14901112723277580*n^30 + 327062325560360*n^31 + 22429224033778*n^32 + 366602803600*n^33 + 23094295940*n^34 + 264617940*n^35 + 15377517*n^36 + 110200*n^37 + 5930*n^38 + 20*n^39 + n^40).
G.f.: 2*x*(1 + 20*x + 400*x^2 + 3800*x^3 + 36100*x^4 + 216600*x^5 + 1299600*x^6 + 5523300*x^7 + 23474025*x^8 + 75116880*x^9 + 240374016*x^10 + 600935040*x^11 + 1502337600*x^12 + 3004675200*x^13 + 6009350400*x^14 + 9765194400*x^15 + 15868440900*x^16 + 21157921200*x^17 + 28210561600*x^18 + 31031617760*x^19 + 34134779536*x^20 + 31031617760*x^21 + 28210561600*x^22 + 21157921200*x^23 + 15868440900*x^24 + 9765194400*x^25 + 6009350400*x^26 + 3004675200*x^27 + 1502337600*x^28 + 600935040*x^29 + 240374016*x^30 + 75116880*x^31 + 23474025*x^32 + 5523300*x^33 + 1299600*x^34 + 216600*x^35 + 36100*x^36 + 3800*x^37 + 400*x^38 + 20*x^39 + x^40)/(1-x)^43. (End)
MATHEMATICA
A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
Table[A157070[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) )
[A157070(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 22 2009
STATUS
approved