A279358 -id:A279358 - 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}).

A300452 Logarithmic transform of the cubes A000578.

Original entry on oeis.org

0, 1, 7, 5, -146, -351, 9936, 51421, -1394000, -12844287, 328407400, 4874111901, -115361217696, -2607873466511, 55768370301112, 1866984952934445, -34886452149332864, -1720211491314549375, 26716801597874981064, 1979492625918149729437, -23490293022369696366560, -2777285149336544358953679

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			E.g.f.: A(x) = x/1! + 7*x^2/2! + 5*x^3/3! - 146*x^4/4! - 351*x^5/5! + 9936*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..408
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Logarithmic Transform
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A009306, A033464, A279358.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i^3)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

nmax = 21; CoefficientList[Series[Log[1 + Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: log(1 + exp(x)*x*(1 + 3*x + x^2)).

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).

Original entry on oeis.org

1, 1, 10, 81, 976, 14505, 258456, 5377897, 127852096, 3419620209, 101625743080, 3322169384721, 118475520287136, 4577175039397753, 190436902905933880, 8489222610046324665, 403657900923994965376, 20393319895130130117729, 1090902632352025316904648

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A000578, A006153, A144109, A279358, A302870, A308861.

nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ Michel Marcus, Mar 10 2022

E.g.f.: 1 / (1 - Sum_{k>=1} k^3*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 7*r + 6*r^2 + r^3)), where r = 0.33649177041401456061485914122406146158245451810028937972189... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jun 29 2019

A281231 Exponential transform of the tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 5, 23, 133, 916, 7107, 61286, 580505, 5968400, 66032901, 780962524, 9817927385, 130572957724, 1829676460991, 26919714974436, 414591408939313, 6665930432840304, 111624874150941193, 1942675652654112012, 35071252458352443001, 655641049733709757516

Offset: 0

Ilya Gutkovskiy, Jan 18 2017

nonn

			E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 23*x^3/3! + 133*x^4/4! + 916*x^5/5! + 7107*x^6/6! + ...

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Exponential Transform
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A279361, A279358.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*j*(j+1)*(j+2)/6, j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 18 2017

Range[0, 21]! CoefficientList[Series[Exp[Exp[x] x (1 + x + x^2/6)], {x, 0, 21}], x]

E.g.f.: exp(exp(x)*x*(1+x+x^2/6)).

A336184 a(n) = n^3 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^3.

Original entry on oeis.org

1, 9, 53, 466, 5569, 82656, 1474045, 30664656, 729036801, 19499288680, 579487528861, 18943592776032, 675568129695601, 26099852672860344, 1085904530481561645, 48407032164910589056, 2301727955153266523521, 116286277045753464506568, 6220517619913795356269725

Offset: 1

Ilya Gutkovskiy, Jul 10 2020

nonn

Cf. A000578, A279358, A300452, A305990, A308862, A336183.

a[n_] := a[n] = n^3 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^3, {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]
nmax = 19; CoefficientList[Series[-Log[1 - Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

E.g.f.: -log(1 - exp(x) * x * (1 + 3*x + x^2)).
E.g.f.: -log(1 - Sum_{k>=1} k^3 * x^k / k!).
a(n) ~ (n-1)! / r^n, where r = 0.336491770414014560614859141224061461582454518... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jul 11 2020

A337826 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).

Original entry on oeis.org

1, 1, 10, 105, 2248, 62445, 2390436, 116650177, 7043659904, 514744959321, 44534754680500, 4493090921151261, 521600149636044480, 68900819660071184149, 10259571068808850618480, 1708054303772376318547125, 315688007001129064574027776, 64370788231256983836207599153

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

nonn

Cf. A023998, A279358, A336227, A337591, A337825.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^4 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} n^3 * x^n / (n!)^2).

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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A300452 Logarithmic transform of the cubes A000578.

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A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).

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A281231 Exponential transform of the tetrahedral numbers (A000292).

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A336184 a(n) = n^3 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^3.

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A337826 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).

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Mathematica

Formula

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).