A300452 -id:A300452 - cp's OEIS frontend

A300455 Logarithmic transform of the triangular numbers A000217.

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}]

A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * k * a(k).

Original entry on oeis.org

0, 1, 6, -33, -512, 19405, 181116, -45817541, 771776384, 280415588121, -23151651942500, -3217963989270569, 816268626535923936, 38087192839910816485, -43268389662374707851552, 2822720920753640236252875, 3297662826737476255127428096, -833876355494162903256716734927

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

sign

Cf. A002190, A101981, A300452, A337590, A337824, A337826.

a[0] = 0; a[n_] := a[n] = n^3 - (1/n) * Sum[Binomial[n, k]^2 (n - k)^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Log[1 + 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 = log(1 + x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).

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

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

cp's OEIS Frontend

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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A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * 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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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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A306325 Expansion of e.g.f. log(1 + exp(x)x(1 + 7x + 6x^2 + x^3)).