OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..1637
FORMULA
a(n) = Sum_{k=0..floor(n/4)} binomial(n+8*k+3,n-4*k) * binomial(4*k,k) / (3*k+1).
D-finite with recurrence: 12*(3*n + 20)*(n + 4)*(3*n + 4)*a(n) - 6*(3*n + 7)*(84*n^2 + 965*n + 2670)*a(n + 1) + 3*(3*n + 10)*(1083*n^2 + 13283*n + 39934)*a(n + 2) - 15*(3*n + 13)*(849*n^2 + 11018*n + 35432)*a(n + 3) + 15*(3*n + 16)*(2247*n^2 + 30541*n + 103437)*a(n + 4) - 39*(3*n + 19)*(1617*n^2 + 22708*n + 79475)*a(n + 5) + 39*(3*n + 22)*(2163*n^2 + 30695*n + 107600)*a(n + 6) - 429*(3*n + 25)*(183*n^2 + 2498*n + 8094)*a(n + 7) + (134111*n^3 + 2697429*n^2 + 15841363*n + 21910860)*a(n + 8) - (17257*n^3 - 346407*n^2 - 11608015*n - 63912471)*a(n + 9) - (42727*n^3 + 1848609*n^2 + 24322580*n + 100417788)*a(n + 10) + (40489*n^3 + 1508433*n^2 + 18031706*n + 68896692)*a(n + 11) - (16943*n^3 + 572325*n^2 + 5966257*n + 17965560)*a(n + 12) + (1201*n^3 - 13995*n^2 - 1186381*n - 10593105)*a(n + 13) + 15*(3*n + 46)*(57*n^2 + 2141*n + 19372)*a(n + 14) - 3*(3*n + 49)*(147*n^2 + 4898*n + 40610)*a(n + 15) + 3*(3*n + 52)*(33*n^2 + 1075*n + 8745)*a(n + 16) - 3*(3*n + 55)*(3*n + 47)*(n + 17)*a(n + 17) = 0. - Robert Israel, May 01 2026
MAPLE
f:= gfun:-rectoproc({(108*n^3 + 1296*n^2 + 4416*n + 3840)*a(n) + (-1512*n^3 - 20898*n^2 - 88590*n - 112140)*a(n + 1) + (9747*n^3 + 152037*n^2 + 757896*n + 1198020)*a(n + 2) + (-38205*n^3 - 661365*n^2 - 3742950*n - 6909240)*a(n + 3) + (101115*n^3 + 1913625*n^2 + 11984505*n + 24824880)*a(n + 4) + (-189189*n^3 - 3855033*n^2 - 26125203*n - 58890975)*a(n + 5) + (253071*n^3 + 5447169*n^2 + 38925510*n + 92320800)*a(n + 6) + (-235521*n^3 - 5177601*n^2 - 37208028*n - 86808150)*a(n + 7) + (134111*n^3 + 2697429*n^2 + 15841363*n + 21910860)*a(n + 8) + (-17257*n^3 + 346407*n^2 + 11608015*n + 63912471)*a(n + 9) + (-42727*n^3 - 1848609*n^2 - 24322580*n - 100417788)*a(n + 10) + (40489*n^3 + 1508433*n^2 + 18031706*n + 68896692)*a(n + 11) + (-16943*n^3 - 572325*n^2 - 5966257*n - 17965560)*a(n + 12) + (1201*n^3 - 13995*n^2 - 1186381*n - 10593105)*a(n + 13) + (2565*n^3 + 135675*n^2 + 2349030*n + 13366680)*a(n + 14) + (-1323*n^3 - 65691*n^2 - 1085496*n - 5969670)*a(n + 15) + (297*n^3 + 14823*n^2 + 246405*n + 1364220)*a(n + 16) + (-27*n^3 - 1377*n^2 - 23361*n - 131835)*a(n + 17), a(0) = 1, a(1) = 4, a(2) = 10, a(3) = 20, a(4) = 36, a(5) = 72, a(6) = 220, a(7) = 936, a(8) = 4045, a(9) = 15836, a(10) = 56174, a(11) = 187148, a(12) = 616651, a(13) = 2114448, a(14) = 7717752, a(15) = 29498000, a(16) = 114243269}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 01 2026
PROG
(PARI) a(n) = sum(k=0, n\4, binomial(n+8*k+3, n-4*k)*binomial(4*k, k)/(3*k+1));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 29 2024
STATUS
approved