A033464 -id:A033464 - cp's OEIS frontend

A300452 Logarithmic transform of the cubes A000578.

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

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			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)).

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177

Offset: 1

Johannes W. Meijer, Jul 27 2016

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The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 1..451
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 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 W. Weisstein MathWorld, Logarithmic Transform.

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.

a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)

N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).

E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A300455 Logarithmic transform of the triangular numbers A000217.

Original entry on oeis.org

0, 1, 2, -1, -11, 19, 201, -764, -7426, 52137, 448435, -5377604, -38712486, 777663613, 4258812299, -149524753650, -505685566184, 36733876797025, 30910872539763, -11174584391207360, 25170998506744790, 4101787001153848461, -24862093152821214653, -1776483826032814964966

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

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			E.g.f.: A(x) = x/1! + 2*x^2/2! - x^3/3! - 11*x^4/4! + 19*x^5/5! + 201*x^6/6! - 764*x^7/7! - 7426*x^8/8! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450
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, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A009306, A033464, A279361, A300452.

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*(i+1)/2)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

nmax = 23; CoefficientList[Series[Log[1 + Exp[x] x (x + 2)/2], {x, 0, nmax}], x] Range[0, nmax]!

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

A306325 Expansion of e.g.f. log(1 + exp(x)x(1 + 7x + 6x^2 + x^3)).

Original entry on oeis.org

0, 1, 15, 35, -650, -5251, 83376, 1623439, -19261584, -836109351, 5365104400, 636771444011, 561938325312, -661384866976523, -7128491581221360, 879709224738485415, 21742632225425026816, -1413667730904479933647, -64871991410092201623024, 2556051301724027073500035, 212244727356899863738042560

Offset: 0

Ilya Gutkovskiy, Feb 07 2019

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Seiichi Manyama, Table of n, a(n) for n = 0..384

Cf. A000583, A033464, A279637, A300452.

a:=series(log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[Log[1 + Exp[x] x (1 + 7 x + 6 x^2 + x^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n^4 - Sum[Binomial[n, k] (n - k)^4 k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 20}]

E.g.f.: log(1 + Sum_{k>=1} k^4*x^k/k!).

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

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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}]

A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

Original entry on oeis.org

0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

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Robert Israel, Table of n, a(n) for n = 0..257

Cf. A002190, A033464, A101981, A337590, A337591, A337825.

S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):
0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # Robert Israel, Jan 07 2024

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

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

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

A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, -1, 0, 1, 7, -1, -2, 0, 1, 15, 5, -26, 9, 0, 1, 31, 35, -146, 29, 6, 0, 1, 63, 149, -650, -351, 756, -155, 0, 1, 127, 539, -2642, -5251, 9936, -1793, 232, 0, 1, 255, 1805, -10346, -46071, 83376, 51421, -45744, 3969, 0

Offset: 1

Seiichi Manyama, May 30 2019

			Square array begins:
   1,    1,     1,     1,       1,        1, ...
   0,    1,     3,     7,      15,       31, ...
   0,   -1,    -1,     5,      35,      149, ...
   0,   -2,   -26,  -146,    -650,    -2642, ...
   0,    9,    29,  -351,   -5251,   -46071, ...
   0,    6,   756,  9936,   83376,   559656, ...
   0, -155, -1793, 51421, 1623439, 28735405, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..4 give A000007(n-1), A009306, A033464, A300452, A306325.

A(n,n) gives A320939.

T[n_, k_] := T[n, k] = n^k - Sum[Binomial[n-1,j] * j^k * T[n-j,k], {j,1,n-1}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A(n,k) = n^k - Sum_{j=1..n-1} binomial(n-1,j)*j^k*A(n-j,k).

A320255 a(n) = n! * [x^n] log(1 + exp(x)(x + (n/2 - 1)x^2)).

Original entry on oeis.org

0, 1, 1, -1, -26, 39, 3666, -7400, -1488416, 3802113, 1322570530, -4095154284, -2187371499312, 7964242253473, 6052757424558586, -25343867475914910, -25988018018090461664, 123032891453320498449, 163684285184147641156098, -864557405968781387651984, -1448111703094244548802632160

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

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

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

Cf. A009306, A033464, A300455, A318118, A318124, A320254.

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

A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.

Original entry on oeis.org

1, 5, 23, 154, 1389, 15636, 211231, 3329264, 59969097, 1215233380, 27362096211, 677690995488, 18310602210445, 535964033279780, 16894811428737495, 570603293774677696, 20556251540382371217, 786832900592755991364, 31889277719673937849243, 1364231113649221829763200

Offset: 1

Ilya Gutkovskiy, Jul 10 2020

nonn

Cf. A000290, A033462, A033464, A305990, A308861, A336184.

a[n_] := a[n] = n^2 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

E.g.f.: -log(1 - exp(x) * x * (1 + x)).
E.g.f.: -log(1 - Sum_{k>=1} k^2 * x^k / k!).
a(n) ~ (n-1)! / r^n, where r = A201941 = 0.444130228823966590585466329490984667... is the root of the equation exp(r)*r*(1+r) = 1. - Vaclav Kotesovec, Jul 11 2020

cp's OEIS Frontend

A300452 Logarithmic transform of the cubes A000578.

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A274805 The logarithmic transform of sigma(n).

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A300455 Logarithmic transform of the triangular numbers A000217.

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A306325 Expansion of e.g.f. log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)).

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

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A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

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A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).

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A320255 a(n) = n! * [x^n] log(1 + exp(x)*(x + (n/2 - 1)*x^2)).

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A306325 Expansion of e.g.f. log(1 + exp(x)x(1 + 7x + 6x^2 + x^3)).

A320255 a(n) = n! * [x^n] log(1 + exp(x)(x + (n/2 - 1)x^2)).