A364623
G.f. satisfies A(x) = 1/(1-x)^3 + x*A(x)^3.
Original entry on oeis.org
1, 4, 18, 112, 847, 7086, 62974, 583002, 5560323, 54249583, 538873135, 5431177821, 55402340842, 570899082760, 5933922697380, 62138800690564, 654949976467593, 6942859160218698, 73972792893687427, 791722414873487767, 8508265804914763731
Offset: 0
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a(n) = sum(k=0, n, binomial(n+5*k+2, 6*k+2)*binomial(3*k, k)/(2*k+1));
A366184
G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x)^3.
Original entry on oeis.org
1, 4, 21, 163, 1487, 14697, 153226, 1659338, 18483960, 210437161, 2437721418, 28640748192, 340473075541, 4087735789616, 49494986770104, 603699827411356, 7410709463933414, 91484338902961485, 1135029142529785303, 14145212892466682781, 176993823220824229047
Offset: 0
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a(n) = sum(k=0, n, binomial(n+8*k+2, n-k)*binomial(3*k, k)/(2*k+1));
A363816
G.f. satisfies A(x) = (1 + x/A(x))/(1 - x)^2.
Original entry on oeis.org
1, 3, 2, 8, -9, 62, -230, 1054, -4753, 22208, -105419, 508396, -2482284, 12248430, -60980860, 305955372, -1545397447, 7852100312, -40105277621, 205798130624, -1060467961487, 5485199090834, -28469067353663, 148220323891484, -773892318396664, 4051261817405034
Offset: 0
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A363816[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2(k-1),n-k]/(2k-1),{k,0,n}];Array[A363816,30,0] (* Paolo Xausa, Oct 20 2023 *)
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a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(2*(k-1), n-k)/(2*k-1));
A364630
G.f. satisfies A(x) = (1+x*A(x)^4)/(1-x)^2.
Original entry on oeis.org
1, 3, 17, 153, 1621, 18732, 229103, 2915498, 38204497, 512027945, 6985933889, 96705749625, 1354868839933, 19175008086962, 273731258980839, 3936883123412972, 56991044183321197, 829750943505927435, 12142121554514962205, 178488780583916045949
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+7*k+1, 8*k+1)*binomial(4*k, k)/(3*k+1));
A363818
G.f. satisfies A(x) = (1 + x/A(x)^2)/(1 - x)^2.
Original entry on oeis.org
1, 3, -1, 24, -125, 924, -6895, 54181, -438737, 3639655, -30769033, 264122781, -2296010693, 20171456222, -178818115155, 1597550237324, -14369097515939, 130010781029079, -1182520161325459, 10806114831458755, -99163805247182631, 913441732959868748
Offset: 0
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a(n) = (-1)^(n-1)*sum(k=0, n, binomial(3*k-1, k)*binomial(2*(2*k-1), n-k)/(3*k-1));
A366179
G.f. A(x) satisfies A(x) = 1/(1 - x)^2 + x*A(x)^3/(1 - x).
Original entry on oeis.org
1, 3, 13, 80, 582, 4627, 38906, 340138, 3060404, 28151835, 263546436, 2502686416, 24048985907, 233410500126, 2284790496700, 22530585455108, 223610524426654, 2231886642819974, 22389017726854323, 225604735477075272, 2282518274913713101
Offset: 0
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a(n) = sum(k=0, n, binomial(n+4*k+1, n-k)*binomial(3*k, k)/(2*k+1));
A366180
G.f. A(x) satisfies A(x) = 1/(1 - x)^2 + x*A(x)^3/(1 - x)^3.
Original entry on oeis.org
1, 3, 15, 109, 909, 8184, 77626, 764226, 7735878, 80011063, 841875232, 8983175079, 96977392945, 1057262750608, 11623867926024, 128730566729686, 1434752590885174, 16080839356274157, 181135636330594960, 2049430159361529977, 23280997677471432102
Offset: 0
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a(n) = sum(k=0, n, binomial(n+6*k+1, n-k)*binomial(3*k, k)/(2*k+1));
Showing 1-7 of 7 results.