A033462 -id:A033462 - cp's OEIS frontend

A274804 The exponential transform of sigma(n).

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[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 exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(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 = 0..531
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, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(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=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

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

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.

Original entry on oeis.org

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}).

A033464 Logarithmic (or "LOG") transform of squares A000290.

Original entry on oeis.org

1, 3, -1, -26, 29, 756, -1793, -45744, 189513, 4700260, -30515629, -730341600, 6948349069, 159130156836, -2123506814505, -46081244842304, 838034409016721, 17029766318842692, -414549408916313189, -7774211453384941440, 251026027696302116181, 4263756050277024153028

Offset: 0

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 0..435

Cf. A000290, A033462.

logtr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end; n->b(n+1) end: a:= logtr(n-> n^2): seq(a(n), n=0..25); # Alois P. Heinz, Sep 14 2008

With[{nn=30},CoefficientList[Series[(Exp[x](1+3x+x^2))/(1+Exp[x]x(1+x)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 03 2019 *)

E.g.f.: exp(x)*(1 + 3*x + x^2)/(1 + exp(x)*x*(1 + x)). - Ilya Gutkovskiy, Mar 06 2018

A279361 Exponential transform of the triangular numbers.

Original entry on oeis.org

1, 1, 4, 16, 80, 471, 3127, 23059, 186468, 1635265, 15422471, 155388399, 1663294756, 18826525771, 224434810797, 2808247979611, 36770685485408, 502505495269521, 7150461569849395, 105723461155720879, 1621191824611307436, 25738508587975433251

Offset: 0

Ilya Gutkovskiy, Dec 10 2016

nonn

			E.g.f.: A(x) = 1 + x/1! + 4*x^2/2! + 16*x^3/3! + 80*x^4/4! + 471*x^5/5! + 3127*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..519
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, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A033462.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j*(j+1)/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 11 2016

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] x (x + 2)/2], {x, 0, 23}], x]

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

A279358 Exponential transform of the cubes A000578.

Original entry on oeis.org

1, 1, 9, 52, 413, 3916, 41077, 481384, 6198425, 86430160, 1296040841, 20763245944, 353272341061, 6353672109760, 120315348389069, 2390488408994536, 49682962883210033, 1077292416660660736, 24313317132393295633, 569937590287796925784, 13850459183086300341341

Offset: 0

Ilya Gutkovskiy, Dec 10 2016

nonn

			E.g.f.: A(x) = 1 + x/1! + 9*x^2/2! + 52*x^3/3! + 413*x^4/4! + 3916*x^5/5! + 41077*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..479
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, Cubic Number

Cf. A000578, A033462.

Column k=3 of A279636.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^3*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 11 2016

Range[0, 20]! CoefficientList[Series[Exp[Exp[x] (x + 3 x^2 + x^3)], {x, 0, 20}], x]

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

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

Original entry on oeis.org

1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..396

Cf. A000290, A006153, A033453, A033462, A302189, A308862.

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

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

E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - Vaclav Kotesovec, Jun 29 2019

A337591 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^3 * a(n-k).

Original entry on oeis.org

1, 1, 6, 51, 760, 15545, 428256, 15043483, 653049664, 34204348305, 2118834917200, 152834879685851, 12670536337934256, 1194143629239156505, 126753440317516749660, 15031687739886065433375, 1977667235694725269563136, 286890421090357737699794209, 45637300134026406622214264592

Offset: 0

Ilya Gutkovskiy, Sep 02 2020

nonn

Cf. A033462, A336227.

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

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

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}]

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

A336961 Expansion of e.g.f. exp(x * (2 + x) * exp(x)).

Original entry on oeis.org

1, 2, 10, 56, 384, 3022, 26626, 258624, 2734360, 31168682, 380196414, 4932536908, 67717987948, 979613124414, 14877703575130, 236469561581768, 3922587278751504, 67743812585483218, 1215417753459838198, 22609895367286957572, 435341977596130683316

Offset: 0

Ilya Gutkovskiy, Aug 09 2020

nonn

Exponential transform of the oblong numbers (A002378).

Cf. A000248, A002378, A033462, A209801, A216507, A279361, A336960.

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

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

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

cp's OEIS Frontend

A274804 The exponential transform of sigma(n).

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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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A033464 Logarithmic (or "LOG") transform of squares A000290.

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A279361 Exponential transform of the triangular numbers.

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A279358 Exponential transform of the cubes A000578.

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A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

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A337591 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^3 * a(n-k).

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

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A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

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