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.

Showing 1-5 of 5 results.

A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

Original entry on oeis.org

1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

Links

Crossrefs

Column k=3 of A309386.
Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.

Programs

  • Maple
    a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)

Formula

a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018

A318179 Expansion of e.g.f. exp((1 - exp(-4*x))/4).

Original entry on oeis.org

1, 1, -3, 5, 25, -343, 2133, -3603, -112975, 1938897, -18008275, 55198805, 1753746377, -45801271943, 649021707397, -4682002329795, -50792700319903, 2692784088681889, -59182401177647011, 801759226622986917, -2169423359710146183, -263145142263538606519, 9869607872225170545333
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

Links

Crossrefs

Column k=4 of A309386.
Cf. A004213, A007696, A009235, A014182, A317996, A318180, A318181.

Programs

  • Maple
    seq(n!*coeff(series(exp((1-exp(-4*x))/4),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[(1 - Exp[-4 x])/4], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-4)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = Sum[(-4)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Table[(-4)^n BellB[n, -1/4], {n, 0, 22}] (* Peter Luschny, Aug 20 2018 *)

Formula

a(n) = Sum_{k=0..n} (-4)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-4)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-4)^n*BellPolynomial_n(-1/4). - Peter Luschny, Aug 20 2018

A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 1, 0, 1, -4, 6, 2, 1, 0, 1, -5, 12, -3, -6, -2, 0, 1, -6, 20, -20, -21, -14, -9, 0, 1, -7, 30, -55, -20, 24, 26, -9, 0, 1, -8, 42, -114, 45, 172, 195, 178, 50, 0, 1, -9, 56, -203, 246, 370, 108, -111, 90, 267, 0, 1, -10, 72, -328, 679, 318, -1105, -2388, -3072, -2382, 413, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2017

Keywords

Examples

			Square array begins:
   1,  1,   1,   1,   1,     1,     1, ...
   0, -1,  -2,  -3,  -4,    -5,    -6, ...
   0,  0,   2,   6,  12,    20,    30, ...
   0,  1,   2,  -3, -20,   -55,  -114, ...
   0,  1,  -6, -21, -20,    45,   246, ...
   0, -2, -14,  24, 172,   370,   318, ...
   0, -9,  26, 195, 108, -1105, -4074, ...

Links

Crossrefs

Columns k=0..4 give A000007, A000587, A213170, A309084, A309085.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292866.
Cf. A292860, A309386.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017
  • Mathematica
    A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 10 2021 *)
A292861[n_, k_] := BellB[k, k - n];
Table[A292861[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

Formula

A(0,k) = 1 and A(n,k) = -k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, -k). - Peter Luschny, Dec 23 2021

A306245 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} k^j * binomial(n-1,j) * A(j,k) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 15, 1, 1, 1, 5, 43, 179, 52, 1, 1, 1, 6, 89, 1279, 3489, 203, 1, 1, 1, 7, 161, 5949, 108472, 127459, 877, 1, 1, 1, 8, 265, 20591, 1546225, 26888677, 8873137, 4140, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 28 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,      1,       1,        1, ...
   1,  1,    1,      1,       1,        1, ...
   1,  2,    3,      4,       5,        6, ...
   1,  5,   17,     43,      89,      161, ...
   1, 15,  179,   1279,    5949,    20591, ...
   1, 52, 3489, 108472, 1546225, 12950796, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000110, A126443, A355081, A355082.
Rows n=0+1, 2 give A000012, A000027(n+1).
Main diagonal gives A309401.
Cf. A309386.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1,
      add(k^j*binomial(n-1, j)*A(j, k), j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jul 28 2019
  • Mathematica
    A[0, _] = 1;
A[n_, k_] := A[n, k] = Sum[k^j Binomial[n-1, j] A[j, k], {j, 0, n-1}];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 29 2020 *)

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(k * x / (1 - x)) / (1 - x). - Seiichi Manyama, Jun 18 2022

A334192 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (k*j + 1)^n / ((-k)^j * j!).

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2020

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,    1,    1,  ...
   0,   0,    0,    0,    0,    0,  ...
  -1,  -2,   -3,   -4,   -5,   -6,  ...
  -1,  -4,   -9,  -16,  -25,  -36,  ...
   2,   4,    0,  -16,  -50, -108,  ...
   9,  64,  189,  384,  625,  864,  ...

Crossrefs

Columns k=1..3 give A293037, A334190, A334191.
Cf. A309386, A334165, A334193 (diagonal).

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (1 - Exp[k x])/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (-x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).
E.g.f. of column k: exp(x + (1 - exp(k*x)) / k).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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