A336950
E.g.f.: 1 / (1 - x * exp(2*x)).
Original entry on oeis.org
1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016
Offset: 0
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nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
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seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020
A336951
E.g.f.: 1 / (1 - x * exp(3*x)).
Original entry on oeis.org
1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020
A351790
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))^j/j!.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 4, 6, 8, 10, 12, ...
6, 21, 42, 69, 102, 141, ...
24, 148, 392, 780, 1336, 2084, ...
120, 1305, 4600, 11145, 22200, 39145, ...
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T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
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T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
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T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));
A351793
Expansion of e.g.f. 1/(1 - x*exp(-4*x)).
Original entry on oeis.org
1, 1, -6, 6, 248, -2120, -12144, 458416, -2194560, -102238848, 2116494080, 12999644416, -1291721856000, 14270887521280, 650218659514368, -24515781088389120, -89087389799317504, 27917287109308284928, -556978307357438705664, -23150337968775391281152
Offset: 0
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a[0] = 1; a[n_] := n!*Sum[(-4*(n - k))^k/k!, {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 19 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-4*x))))
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a(n) = n!*sum(k=0, n, (-4*(n-k))^k/k!);
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a(n) = if(n==0, 1, n*sum(k=0, n-1, (-4)^(n-1-k)*binomial(n-1, k)*a(k)));
Showing 1-4 of 4 results.