A351761
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 4, 12, 24, 40, 60, ...
0, 21, 102, 279, 588, 1065, ...
0, 148, 1160, 4332, 11536, 25220, ...
0, 1305, 16490, 84075, 282900, 746525, ...
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T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);
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T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
A351778
Expansion of e.g.f. 1/(1 + 3*x*exp(x)).
Original entry on oeis.org
1, -3, 12, -63, 420, -3435, 33462, -379155, 4903896, -71318259, 1152202290, -20474486043, 396890715636, -8334602179995, 188486823883134, -4567087352339235, 118039115079323952, -3241465018561379427, 94249758656850366186, -2892678859033260044043
Offset: 0
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With[{nn=30},CoefficientList[Series[1/(1+3x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 21 2024 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*exp(x))))
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a(n) = n!*sum(k=0, n, (-3)^(n-k)*(n-k)^k/k!);
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a(n) = if(n==0, 1, -3*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));
A355501
Expansion of e.g.f. exp(3 * x * exp(x)).
Original entry on oeis.org
1, 3, 15, 90, 633, 5028, 44217, 424434, 4399953, 48858984, 577372809, 7221983838, 95192539641, 1317190650636, 19071213218745, 288112248054882, 4530217559806497, 73976635012027344, 1252091246140278153, 21926952634345281030, 396671314081806278601
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[3x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 23 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*exp(x))))
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*x)^k/(1-k*x)^(k+1)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, j*binomial(i-1, j-1)*v[i-j+1])); v;
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a(n) = sum(k=0, n, 3^k*k^(n-k)*binomial(n, k));
Showing 1-3 of 3 results.