cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A308497 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} binomial(j+k-1,k) * x^j/j).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 8, 1, 4, 10, 15, 24, 26, 1, 5, 17, 34, 54, 120, 194, 1, 6, 26, 69, 104, 240, 720, 1142, 1, 7, 37, 126, 204, 200, 1350, 5040, 9736, 1, 8, 50, 211, 408, -330, -400, 9450, 40320, 81384, 1, 9, 65, 330, 794, -1704, -12510, -2800, 78120, 362880, 823392
Offset: 1

Views

Author

Seiichi Manyama, Jun 01 2019

Keywords

Comments

Column k > 2 is asymptotic to -2*(n-1)! * cos(n*arctan(sin(Pi/k)/(cos(Pi/k) - (k-1)^(1/k)))) / (1 + 1/(k-1)^(2/k) - 2*cos(Pi/k)/(k-1)^(1/k))^(n/2). - Vaclav Kotesovec, May 12 2021

Examples

			Square array begins:
     1,   1,    1,    1,      1,      1, ...
     0,   1,    2,    3,      4,      5, ...
     1,   2,    5,   10,     17,     26, ...
     1,   6,   15,   34,     69,    126, ...
     8,  24,   54,  104,    204,    408, ...
    26, 120,  240,  200,   -330,  -1704, ...
   194, 720, 1350, -400, -12510, -51696, ...

Links

Crossrefs

Columns k=0..5 give A089064, A000142(n-1), (-1)^(n+1) * A009383(n), A308499, A344217, A344218.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = ((n+k-1)! - Sum[Binomial[n-1,j] * (j+k-1)! * T[n-j,k], {j,1,n-1}])/k!; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Formula

A(n,k) = (1/k!) * ((n+k-1)! - Sum_{j=1..n-1} binomial(n-1,j) * (j+k-1)! * A(n-j,k)).
E.g.f.: log(1 + (1/(1-x)^k - 1)/k). - Vaclav Kotesovec, May 12 2021

A079642 Matrix product of unsigned Stirling1-triangle |A008275(n,k)| and Stirling1-triangle A008275(n,k).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 8, 5, 10, 0, 1, 26, 58, 15, 20, 0, 1, 194, 217, 238, 35, 35, 0, 1, 1142, 2035, 1008, 728, 70, 56, 0, 1, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1, 81384, 134164, 85410, 47815, 9660, 4116, 210, 120, 0, 1, 823392, 1243770, 983059
Offset: 1

Views

Author

Vladeta Jovovic, Jan 30 2003

Keywords

Comments

Also the Bell transform of A089064(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Examples

			1; 0,1; 1,0,1; 1,4,0,1; 8,5,10,0,1; 26,58,15,20,0,1; ...

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add((-1)^n*(k-1)!*combinat:-stirling1(n+1, k), k=1..n+1), 9); # Peter Luschny, Jan 26 2016
  • Mathematica
    BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, Sum[(-1)^n*(k-1)! StirlingS1[n+1, k], {k, 1, n+1} ] ], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

T(n, k) = Sum_{i=k..n} |A008275(n, i)| * A008275(i, k).
E.g.f.: (1-log(1-x))^y. - Vladeta Jovovic, Nov 22 2003

A330388 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * log(1 + x)^k / (k * (1 - log(1 + x)^k)).

Original entry on oeis.org

1, 0, 7, -37, 338, -2816, 28418, -340334, 5015080, -84244704, 1536606168, -29753884392, 609895549872, -13243687082016, 305507366834832, -7523621131117296, 198844500026698752, -5649686902983730560, 171839087043420258432, -5545292300345590210944
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2019

Keywords

Links

Crossrefs

Cf. A000009, A000593, A008275, A088311, A089064, A265024, A298905, A330354, A330387.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + Log[1 + x]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Formula

E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A298905.
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + log(1 + x)^k). - Vaclav Kotesovec, Dec 15 2019
Conjecture: a(n) ~ n! * (-1)^(n+1) * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

A336440 a(n) = (n!)^n * [x^n] -log(1 + Sum_{k>=1} (-x)^k / k^n).

Original entry on oeis.org

0, 1, 1, 53, 65656, 4306202624, 21250781850448256, 11198392471992778644752768, 847058443993661249394101877997568000, 11916672812223274564264480372420932763474540363776, 39215070895580530235582705162664184972620228444352744200981184512
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2020

Keywords

Crossrefs

Cf. A089064, A336437, A336438, A336439.

Programs

  • Mathematica
    Table[(n!)^n SeriesCoefficient[-Log[1 + Sum[(-x)^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 0, (-1)^(n + 1) ((n - 1)!)^k - (1/n) Sum[(-1)^(n - j) (Binomial[n, j] (n - j - 1)!)^k j b[j, k], {j, 1, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

A325873 T(n, k) = [x^k] Sum_{k=0..n} |Stirling1(n, k)|*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 8, 5, 10, 0, 1, 0, 26, 58, 15, 20, 0, 1, 0, 194, 217, 238, 35, 35, 0, 1, 0, 1142, 2035, 1008, 728, 70, 56, 0, 1, 0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1, 0, 81384, 134164, 85410, 47815, 9660, 4116, 210, 120, 0, 1
Offset: 0

Views

Author

Peter Luschny, Jun 27 2019

Keywords

Examples

			Triangle starts:
[0] [1]
[1] [0,    1]
[2] [0,    0,     1]
[3] [0,    1,     0,     1]
[4] [0,    1,     4,     0,    1]
[5] [0,    8,     5,    10,    0,    1]
[6] [0,   26,    58,    15,   20,    0,   1]
[7] [0,  194,   217,   238,   35,   35,   0,  1]
[8] [0, 1142,  2035,  1008,  728,   70,  56,  0, 1]
[9] [0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1]

Crossrefs

Columns k=0..2 give A000007, A089064, A341575.
Cf. A079642 (variant), A129062, A325872.

Programs

  • Mathematica
    p[n_] := Sum[Abs[StirlingS1[n, k]] FactorialPower[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten
  • PARI
    T(n, k) = sum(j=k, n, abs(stirling(n, j, 1))*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025
  • Sage
    def a_row(n):
    s = sum(stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..10)]

Formula

From Seiichi Manyama, Apr 18 2025: (Start)
T(n,k) = Sum_{j=k..n} |Stirling1(n,j)| * Stirling1(j,k).
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 - log(1 - x)). (End)

A341575 E.g.f.: log(1 - log(1 - x))^2 / 2.

Original entry on oeis.org

1, 0, 4, 5, 58, 217, 2035, 13470, 134164, 1243770, 14129410, 164244808, 2151576620, 29671566836, 444758323628, 7055358559376, 119546765395744, 2139179551573104, 40486788832168944, 805969129348431936, 16860672502118423136, 369459637224850523808, 8467140450141232328160
Offset: 2

Views

Author

Ilya Gutkovskiy, Feb 15 2021

Keywords

Crossrefs

Cf. A000254, A000558, A008275, A079642, A081048, A089064, A302547, A302548, A341587.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 - Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[Abs[StirlingS1[n, k]] StirlingS1[k, 2], {k, 2, n}], {n, 2, 24}]

Formula

a(n) = Sum_{k=2..n} |Stirling1(n, k)| * Stirling1(k, 2).
a(n) = (-1)^n * Sum_{k=2..n} Stirling1(n, k) * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.

A075792 E.g.f.: 1 + log(1+f(x)) where f(x) = log(1/(2-exp(x))) = e.g.f. for A000629.

Original entry on oeis.org

1, 1, 1, 2, 8, 44, 302, 2512, 24558, 275676, 3493862, 49339784, 768182846, 13071470788, 241332513606, 4804601266896, 102599581877918, 2339270285673068, 56716397892998246, 1457071974028941400, 39538338850995279294, 1130018112921128323668, 33928819073838398622662
Offset: 0

Views

Author

N. J. A. Sloane, Oct 15 2002

Keywords

Comments

Stirling transform of A089064(n)=[1,0,1,1,8,26,...] is a(n)=[1,1,2,8,44,...]. - Michael Somos, Mar 04 2004

Crossrefs

Cf. A000629, A089064.

Programs

  • PARI
    a(n)=if(n<0,0,n!*polcoeff(1+log(1+log(1/(2-exp(x+x*O(x^n))))),n))

A306037 Expansion of e.g.f. 1/(1 + log(1 - log(1 + x))).

Original entry on oeis.org

1, 1, 2, 7, 31, 178, 1200, 9588, 86592, 887086, 10035164, 125472246, 1705102394, 25175822644, 399387494956, 6801042408728, 123348694663480, 2379855020533664, 48569042602254128, 1047134236970183664, 23748242269316806752, 565834452464428045872, 14117321495269290091440
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

Examples

			1/(1 + log(1 - log(1 + x))) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 178*x^5/5! + 1200*x^6/6! + ...

Links

Crossrefs

Cf. A007840, A089064, A305323, A305988.

Programs

  • Maple
    a:=series(1/(1+log(1-log(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + Log[1 - Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[StirlingS1[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 22}]
a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! StirlingS1[k, j], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A007840(k).
a(n) ~ n! * exp(-exp(-1)) / (exp(1 - exp(-1)) - 1)^(n+1). - Vaclav Kotesovec, Jul 01 2018

A306038 Expansion of e.g.f. (1 + x)/(1 - log(1 + x)).

Original entry on oeis.org

1, 2, 3, 5, 12, 34, 122, 482, 2328, 11640, 71952, 424368, 3312240, 21357504, 217045488, 1351338864, 19990187520, 89379824256, 2631270916224, 892036259712, 507945420198144, -3068802187635456, 142961233091051520, -1849617314640322560, 55640352746480440320
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

Examples

			(1 + x)/(1 - log(1 + x)) = 1 + 2*x + 3*x^2/2! + 5*x^3/3! + 12*x^4/4! + 34*x^5/5! + 122*x^6/6! + ...

Links

Crossrefs

Cf. A000522, A006252, A059099, A089064.

Programs

  • Maple
    S:= series((1+x)/(1-log(1+x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jun 19 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[(1 + x)/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
FullSimplify[Table[Sum[StirlingS1[n, k] E Gamma[1 + k, 1], {k, 0, n}], {n, 0, 24}]]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000522(k).

A307125 Expansion of e.g.f. log(1 - log(1 - x*exp(x))).

Original entry on oeis.org

0, 1, 2, 4, 17, 123, 1052, 10568, 125750, 1726189, 26730394, 460982300, 8766443952, 182229703043, 4110207945794, 99970680376908, 2608221938476016, 72656914458625593, 2152355976206481570, 67562405794276542004, 2240111797037473955984, 78229640115171735522015, 2870092624821982184377202
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 26 2019

Keywords

Crossrefs

Cf. A007550, A009444, A089064, A307126.

Programs

  • Maple
    a:=series(log(1-log(1-x*exp(x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Log[1 - Log[1 - x Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x = 'x + O('x^30)); concat(0, Vec(serlaplace(log(1 - log(1 - x*exp(x)))))) \\ Michel Marcus, Mar 26 2019
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