A279644 -id:A279644 - cp's OEIS frontend

A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 10, 15, 1, 1, 9, 22, 41, 52, 1, 1, 17, 52, 125, 196, 203, 1, 1, 33, 130, 413, 836, 1057, 877, 1, 1, 65, 340, 1445, 3916, 6277, 6322, 4140, 1, 1, 129, 922, 5261, 19676, 41077, 52396, 41393, 21147, 1, 1, 257, 2572, 19685, 104116, 288517, 481384, 479593, 293608, 115975

Offset: 0

Alois P. Heinz, Dec 16 2016

			Square array A(n,k) begins:
:   1,    1,    1,     1,      1,       1,        1, ...
:   1,    1,    1,     1,      1,       1,        1, ...
:   2,    3,    5,     9,     17,      33,       65, ...
:   5,   10,   22,    52,    130,     340,      922, ...
:  15,   41,  125,   413,   1445,    5261,    19685, ...
:  52,  196,  836,  3916,  19676,  104116,   572036, ...
: 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Kronecker delta

Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.
Rows n=0+1,2 give: A000012, A000051.
Main diagonal gives A279644.
Cf. A145460.

egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):
A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).

A320939 a(n) = n! * [x^n] log(1 + Sum_{k>=1} k^n*x^k/k!).

Original entry on oeis.org

0, 1, 3, 5, -650, -46071, 3121776, 5538166381, 3146076001776, -10459815889305231, -100694615309371571840, -193538025548431984737219, 38912028315765820944424730112, 2554132880645627969533690819801657, -106074951996903194289368162206783509504

Offset: 0

Ilya Gutkovskiy, Oct 28 2018

sign

a(n) is the n-th term of the logarithmic transform of the n-th powers.

Alois P. Heinz, Table of n, a(n) for n = 0..77
N. J. A. Sloane, Transforms

Cf. A009306, A033464, A279644, A300452.

seq(coeff(series(factorial(n)*log(1+add(k^n*x^k/factorial(k),k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 28 2018

Table[n! SeriesCoefficient[Log[1 + Sum[k^n x^k/k!, {k, 1, n}]], {x, 0, n}], {n, 0, 14}]

A362384 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xz.

Original entry on oeis.org

1, 1, 4, 12, 81, 934, 23703, 1219177

Offset: 0

Andrew Howroyd, Apr 24 2023

Cf. A001329 (magmas), A279644 (labeled case), A362385.

A320254 a(n) = n! * [x^n] exp(exp(x)(x + (n/2 - 1)x^2)).

Original entry on oeis.org

1, 1, 3, 16, 125, 1291, 16177, 241207, 4153193, 81082225, 1770989921, 42763506919, 1131353484637, 32541516492811, 1011058416700529, 33745374949198231, 1204107124715441873, 45741398365345761073, 1843069565594762478145, 78511973999963036415967, 3525468554804288803649381

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

nonn

For n > 2, a(n) is the n-th term of the exponential transform of n-gonal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000248, A033462, A279361, A279644, A318118, A318124, A320255.

Table[n! SeriesCoefficient[Exp[Exp[x] (x + (n/2 - 1) x^2)], {x, 0, n}], {n, 0, 20}]

cp's OEIS Frontend

A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

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A320939 a(n) = n! * [x^n] log(1 + Sum_{k>=1} k^n*x^k/k!).

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A362384 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xz.

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Author

Keywords

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A320254 a(n) = n! * [x^n] exp(exp(x)(x + (n/2 - 1)x^2)).

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cp's OEIS Frontend

A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320939 a(n) = n! * [x^n] log(1 + Sum_{k>=1} k^n*x^k/k!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A362384 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xz.

Views

Author

Keywords

Crossrefs

A320254 a(n) = n! * [x^n] exp(exp(x)*(x + (n/2 - 1)*x^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A320254 a(n) = n! * [x^n] exp(exp(x)(x + (n/2 - 1)x^2)).