A361568
Expansion of e.g.f. exp(x^3/6 * (1+x)^3).
Original entry on oeis.org
1, 0, 0, 1, 12, 60, 130, 420, 8400, 101080, 781200, 4435200, 37714600, 607807200, 8660652000, 94007313400, 914497584000, 11566931376000, 198256136478400, 3275456501116800, 46558791351072000, 636647461257808000, 10238792220969312000, 194852563745775936000
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(1+x)^3)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;
A361569
Expansion of e.g.f. exp(x^4/24 * (1+x)^4).
Original entry on oeis.org
1, 0, 0, 0, 1, 20, 180, 840, 1715, 2520, 88200, 1940400, 29111775, 303603300, 2188286100, 12549537000, 143029511625, 3397035642000, 71419225878000, 1170096883956000, 15075357741068625, 163540869094102500, 2025016641129982500, 40912918773391665000
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*(1+x)^4)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*binomial(4, j-4)*v[i-j+1]/(i-j)!)); v;
A373740
Expansion of e.g.f. exp(x^2/2 * (1 + x)).
Original entry on oeis.org
1, 0, 1, 3, 3, 30, 105, 315, 2625, 11340, 57645, 467775, 2505195, 17027010, 142026885, 922296375, 7493911425, 65886420600, 503693415225, 4625660914875, 43369908657075, 379618464975750, 3824934458169825, 38406952928819475, 376103907454500225
Offset: 0
A373741
Expansion of e.g.f. exp(x^2/2 * (1 + x)^3).
Original entry on oeis.org
1, 0, 1, 9, 39, 150, 1365, 13545, 105945, 918540, 10603845, 127806525, 1468823895, 18253765530, 257397445305, 3770163121725, 55637459903025, 866703333295800, 14468243658093225, 250223925107581425, 4426399346291497575, 81488489549760042750
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[x^2/2 (1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 26 2025 *)
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a(n) = n!*sum(k=0, n\2, binomial(3*k, n-2*k)/(2^k*k!));
A373743
Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).
Original entry on oeis.org
1, 0, 0, 1, 8, 20, 10, 280, 3360, 20440, 67200, 462000, 7407400, 73673600, 482081600, 3364761400, 47311264000, 657536880000, 6586994814400, 58707179731200, 740032028736000, 11832726841936000, 161121297104768000, 1857897194273120000, 23875495204536976000
Offset: 0
Showing 1-5 of 5 results.