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 (41, -820, 10660, -101270, 749398, -4496388, 22481940, -95548245, 350343565, -1121099408, 3159461968, -7898654920, 17620076360, -35240152720, 63432274896, -103077446706, 151584480450, -202112640600, 244662670200, -269128937220, 269128937220, -244662670200, 202112640600, -151584480450, 103077446706, -63432274896, 35240152720, -17620076360, 7898654920, -3159461968, 1121099408, -350343565, 95548245, -22481940, 4496388, -749398, 101270, -10660, 820, -41, 1).
FORMULA
a(n) = T(n,20); 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+19, 20)*Hypergeometric3F2([-19, -n, 1-n], [2, -n-19], 1).
a(n) = (137846528820/40!)*n*(n+1)*(295950609069496384270872084480000000 + 768802633735657375654446366720000000*n + 1329504929585504813849213140992000000*n^2 + 1290742342817244773843889039605760000*n^3 + 1094357439529328748458516078002176000*n^4 + 612766113778575140689735509285273600*n^5 + 334228753141512703020765378377809920*n^6 + 125103295909205358813292403873120256*n^7 + 49218727808847235410751174949228544*n^8 + 13269339361037181895414921845144576*n^9 + 4016584445427935868170163264804864*n^10 + 815165270428049073818572136963328*n^11 + 197974483136507211917478313071872*n^12 + 31108483670185057904409322050688*n^13 + 6244038933930696351877891958272*n^14 + 773683666573321735532607476256*n^15 + 131217385198850594964429765744*n^16 + 12969478215579974430537627276*n^17 + 1890935510804343168840278104*n^18 + 150029328423053669455781465*n^19 + 19066083072333125878657535*n^20 + 1216465853960978843551515*n^21 + 136285407600184771625385*n^22 + 6973959244303571061060*n^23 + 695382022718273834940*n^24 + 28325593615993410660*n^25 + 2534141220949541220*n^26 + 81059848291860174*n^27 + 6552284226337026*n^28 + 160984848978954*n^29 + 11828920639006*n^30 + 215437887572*n^31 + 14466923228*n^32 + 183962712*n^33 + 11343228*n^34 + 89889*n^35 + 5111*n^36 + 19*n^37 + n^38).
G.f.: 2*x*(1 + 19*x + 361*x^2 + 3249*x^3 + 29241*x^4 + 165699*x^5 + 938961*x^6 + 3755844*x^7 + 15023376*x^8 + 45070128*x^9 + 135210384*x^10 + 315490896*x^11 + 736145424*x^12 + 1367127216*x^13 + 2538950544*x^14 + 3808425816*x^15 + 5712638724*x^16 + 6982113996*x^17 + 8533694884*x^18 + 8533694884*x^19 + 8533694884*x^20 + 6982113996*x^21 + 5712638724*x^22 + 3808425816*x^23 + 2538950544*x^24 + 1367127216*x^25 + 736145424*x^26 + 315490896*x^27 + 135210384*x^28 + 45070128*x^29 + 15023376*x^30 + 3755844*x^31 + 938961*x^32 + 165699*x^33 + 29241*x^34 + 3249*x^35 + 361*x^36 + 19*x^37 + x^38)/(1-x)^41. (End)
MATHEMATICA
A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
Table[A157069[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) )
[A157069(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 22 2009
STATUS
approved