A365878
Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^4 ).
Original entry on oeis.org
1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 8552557, 47880020, 270401515, 1539288570, 8821594865, 50860072024, 294774097800, 1716506373521, 10037592274363, 58920231785426, 347051995986538, 2050627029532225, 12151336260368205
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(5*n-k+3, n-k))/(n+1);
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def A365878(n):
h = binomial(5*n + 3, n) * hypergeometric([-n, 3*(n + 1)], [-5 * n - 3], -1) / (n + 1)
return simplify(h)
print([A365878(n) for n in range(24)]) # Peter Luschny, Sep 21 2023
A368079
Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).
Original entry on oeis.org
1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 5258196762, 51426990112, 507537393600, 5048033356128, 50549237164615, 509197913456922, 5156339940802941, 52460340305220466, 535976129228082972, 5496745175387480976
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^2)^3)/x)
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a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
A365880
Expansion of (1/x) * Series_Reversion( x*(1+x)^4*(1-x)^5 ).
Original entry on oeis.org
1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 21163038, 128845344, 790810400, 4894134376, 30486741869, 191068074202, 1203710067455, 7619193325238, 48430121151156, 309011352878208, 1978450305442086, 12706836843595840
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(4*n+k+3, k)*binomial(6*n-k+4, n-k))/(n+1);
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def A365880(n):
h = binomial(6*n + 4, n) * hypergeometric([-n, 4*(n + 1)], [-6 * n - 4], -1) / (n + 1)
return simplify(h)
print([A365880(n) for n in range(23)]) # Peter Luschny, Sep 21 2023
A370270
Coefficient of x^n in the expansion of 1/( (1-x)^2 * (1-x^2)^3 )^n.
Original entry on oeis.org
1, 2, 16, 110, 840, 6502, 51424, 411602, 3326600, 27082460, 221776016, 1824750424, 15073212648, 124926064460, 1038330110400, 8651387371360, 72238476287112, 604327981885262, 5064140053702240, 42500097815152940, 357157266768270840, 3005093769261481238
Offset: 0
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a(n, s=2, t=3, u=2) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
Showing 1-4 of 4 results.