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 1-8 of 8 results.

A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 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,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A242817.
Same array, different indexing is A189233.
Cf. A292861.

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[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {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

A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).

Original entry on oeis.org

1, -1, 2, -3, -20, 370, -4074, 34293, -138312, -2932533, 106271090, -2192834490, 32208497124, -206343936097, -7657279887698, 412496622532785, -12455477719752976, 260294034150380430, -2256541295745391542, -122593550603339550843, 8728842979656718306780
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2017

Keywords

Links

Crossrefs

Main diagonal of A292861.
Cf. A242817.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 25 2017
  • Mathematica
    Table[n!*SeriesCoefficient[E^(n*(1 - E^x)),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 25 2017 *)
a[n_] := BellB[n, -n]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 23 2021 *)
  • PARI
    {a(n) = sum(k=0, n, (-n)^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019
  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << k * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[j]}}
  ary
end
def A292866(n)
  (0..n).map{|i| A(-i, i)[-1]}
end
p A292866(20)

Formula

a(n) = exp(n) * Sum_{k>=0} (-n)^k*k^n/k!. - Ilya Gutkovskiy, Jul 13 2019
a(n) = Sum_{k=0..n} (-n)^k * Stirling2(n,k). - Seiichi Manyama, Jul 28 2019
a(n) = BellPolynomial(n, -n). - Peter Luschny, Dec 23 2021

A350263 Triangle read by rows. T(n, k) = BellPolynomial(n, -k).

Original entry on oeis.org

1, 0, -1, 0, 0, 2, 0, 1, 2, -3, 0, 1, -6, -21, -20, 0, -2, -14, 24, 172, 370, 0, -9, 26, 195, 108, -1105, -4074, 0, -9, 178, -111, -2388, -4805, 2046, 34293, 0, 50, 90, -3072, -3220, 23670, 87510, 111860, -138312, 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533
Offset: 0

Views

Author

Peter Luschny, Dec 23 2021

Keywords

Examples

			[0] 1
[1] 0,  -1
[2] 0,   0,     2
[3] 0,   1,     2,    -3
[4] 0,   1,    -6,   -21,   -20
[5] 0,  -2,   -14,    24,   172,    370
[6] 0,  -9,    26,   195,   108,  -1105, -  4074
[7] 0,  -9,   178,  -111, -2388,  -4805,    2046,    34293
[8] 0,  50,    90, -3072, -3220,  23670,   87510,   111860,  -138312
[9] 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533

Crossrefs

Main diagonal: A292866, column 1: A000587, variant: A292861.

Programs

  • Maple
    A350263 := (n, k) -> ifelse(n = 0, 1, BellB(n, -k)):
seq(seq(A350263(n, k), k = 0..n), n = 0..9);
  • Mathematica
    T[n_, k_] := BellB[n, -k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2020

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...

Links

Crossrefs

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
Main diagonal gives A334241.
Cf. A292861, A335975.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

Formula

T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.

A309386 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2019

Keywords

Examples

			Square array begins:
   1,  1,   1,    1,    1,    1,     1, ...
   1,  1,   1,    1,    1,    1,     1, ...
   1,  0,  -1,   -2,   -3,   -4,    -5, ...
   1, -1,  -1,    1,    5,   11,    19, ...
   1,  1,   9,   19,   25,   21,     1, ...
   1,  2, -23, -128, -343, -674, -1103, ...
   1, -9, -25,  379, 2133, 6551, 15211, ...

Links

Crossrefs

Columns k=0..6 give A000012, (-1)^n * A000587(n), A009235, A317996, A318179, A318180, A318181.
Rows n=0+1, 2 give A000012, A024000.
Main diagonal gives A318183.
Cf. A241578, A292861.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)

Formula

E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.
A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.

A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...

Crossrefs

Columns k=0..3 give A000007, A302397, A351777, A351778.
Main diagonal gives A351779.
Cf. A292861, A351761.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 + k*x*exp(x)).
T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.

Original entry on oeis.org

1, -3, 6, -3, -21, 24, 195, -111, -3072, -4053, 57003, 277854, -697539, -12261567, -29861778, 371727465, 3511027599, 2028432480, -188521156857, -1470389129931, 1655487186864, 121873222577823, 915525253963023, -2095901567014530, -103715912230195863, -836215492271268459
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 11 2019

Keywords

Links

Crossrefs

Column k = 3 of A292861.
Cf. A000587, A027710, A213170, A309085, A317996.

Programs

  • Magma
    [1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // Marius A. Burtea, Jul 27 2019
  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
      (-3)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 17 2022
  • Mathematica
    Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
Table[BellB[n, -3], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

Formula

G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(3*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).

A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

Original entry on oeis.org

1, -4, 12, -20, -20, 172, 108, -2388, -3220, 47532, 161900, -1062740, -8532628, 13623212, 431041132, 1206169260, -17833021588, -169685043796, 180187176044, 13462762665132, 79377664422252, -553096696140884, -11670986989785492, -44371854928405844, 829755609457185644
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 11 2019

Keywords

Links

Crossrefs

Column k = 4 of A292861.
Cf. A000587, A078944, A213170, A309084, A318179.

Programs

  • Magma
    [1] cat [(&+[((-4)^k*StirlingSecond(m,k)):k in [0..m]]):m in [1..24]]; // Marius A. Burtea, Jul 11 2019
  • Mathematica
    Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]
Table[BellB[n, -4], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = sum(k=0, n, (-4)^k * stirling(n,k,2)); \\ Michel Marcus, Jul 12 2019

Formula

G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(4*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).
Showing 1-8 of 8 results.

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