A376320
G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^4.
Original entry on oeis.org
1, 4, 26, 200, 1691, 15180, 142038, 1370076, 13526645, 136024876, 1388394234, 14346699052, 149790104030, 1577765967600, 16745718467070, 178912981116840, 1922688816819276, 20769064846817136, 225384498769815750, 2455985319885345820, 26862562977746930145, 294807644917408047060
Offset: 0
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a(n, s=1, t=4) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^4)/x)
A376328
G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^5.
Original entry on oeis.org
1, 5, 40, 380, 3970, 44051, 509575, 6077435, 74194780, 922644310, 11646083631, 148827827450, 1921724362880, 25034267112600, 328614891689845, 4342322118727300, 57715241768897445, 771087466276360970, 10349495416322497575, 139486475071720234920, 1886980259513934080860, 25613816043115261657425
Offset: 0
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a(n, s=1, t=5) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^5)/x)
Original entry on oeis.org
1, 4, 21, 112, 615, 3432, 19383, 110448, 633726, 3656360, 21191555, 123286440, 719539015, 4210967880, 24702429825, 145210795200, 855172338570, 5044470461352, 29799593861974, 176268499363840, 1043889366927771, 6188748520285584, 36726461258947569, 218146172715259488, 1296812014083995850
Offset: 0
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seq(add(binomial(j,2*j-3*n-4)*binomial(2*n+2,j),j=0...2*n+2),n=0..25); # Mark van Hoeij, May 12 2013
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a(n)=sum(j=0,2*n+2,binomial(j, 2*j-3*n-4)*binomial(2*n+2, j)); \\ Joerg Arndt, May 13 2013~
A027911
a(n) = T(2*n+1,n), with T given by A027907.
Original entry on oeis.org
1, 3, 15, 77, 414, 2277, 12727, 71955, 410346, 2355962, 13599915, 78855339, 458917850, 2679183405, 15683407785, 92022516525, 541050073146, 3186886397310, 18801598011274, 111083331666918, 657153430251396, 3892199032434105, 23077435617920925, 136963282273730613, 813597690808666386
Offset: 0
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seq(add(binomial(j,2*j-2-3*n)*binomial(2*n+1,j),j=0...2*n+1),n=0..20); # Mark van Hoeij, May 12 2013
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Table[GegenbauerC[n, -2 n - 1, -1/2], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
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makelist(ultraspherical(n,-2*n-1,-1/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */
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a(n)=sum(j=0, 2*n+1, binomial(j, 2*j-2-3*n)*binomial(2*n+1, j)); \\ Joerg Arndt, Oct 20 2016
A369506
Expansion of (1/x) * Series_Reversion( x / ((1+x)^3+x^2)^2 ).
Original entry on oeis.org
1, 6, 53, 548, 6192, 74074, 922142, 11822082, 155024190, 2069570934, 28033435791, 384329462490, 5322745393480, 74357950874850, 1046564375245893, 14826433687124098, 211251475010201934, 3025331234242178508, 43523061969049245589, 628692982662691174722
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3+x^2)^2)/x)
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a(n) = sum(k=0, n\2, binomial(2*n+2, k)*binomial(6*n-3*k+6, n-2*k))/(n+1);