A289192
A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880
Offset: 0
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, ...
: 1, 2, 3, 4, 5, 6, ...
: 2, 7, 14, 23, 34, 47, ...
: 6, 34, 86, 168, 286, 446, ...
: 24, 209, 648, 1473, 2840, 4929, ...
: 120, 1546, 5752, 14988, 32344, 61870, ...
Columns k=0-10 give:
A000142,
A002720,
A087912,
A277382,
A289147,
A289211,
A289212,
A289213,
A289214,
A289215,
A289216.
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A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
A[n_, k_] := n! * LaguerreL[n, -k];
Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)
-
{T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021
-
T(n, k) = n!*pollaguerre(n, 0, -k); \\ Michel Marcus, Feb 05 2021
-
from sympy import binomial, factorial as f
def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017
A330260
a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.
Original entry on oeis.org
1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876
Offset: 0
-
[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019
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Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]
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a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019
A341033
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 13, 37, 73, 121, 181, ...
1, 73, 361, 1009, 2161, 3961, ...
1, 501, 4361, 17341, 48081, 108101, ...
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T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)
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{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
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{T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}
A338435
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!*LaguerreL(n, -k*n).
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 14, 6, 1, 4, 34, 168, 24, 1, 5, 62, 654, 2840, 120, 1, 6, 98, 1626, 17688, 61870, 720, 1, 7, 142, 3246, 59928, 616120, 1649232, 5040, 1, 8, 194, 5676, 151064, 2844120, 26252496, 51988748, 40320, 1, 9, 254, 9078, 318744, 9052120, 165100752, 1322624016, 1891712384, 362880
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
2, 14, 34, 62, 98, ...
6, 168, 654, 1626, 3246, ...
24, 2840, 17688, 59928, 151064, ...
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T[n_, k_] := n! * LaguerreL[n, -k*n]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 05 2021 *)
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T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);
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T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ Michel Marcus, Feb 05 2021
Showing 1-4 of 4 results.