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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A367920 Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).

Original entry on oeis.org

1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A126617, A194689, A367888, A367891, A367919, A367921.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

A159830 Exponential Riordan array [exp(exp(x)-1-2x),x].

Original entry on oeis.org

1, -1, 1, 2, -2, 1, -3, 6, -3, 1, 7, -12, 12, -4, 1, -10, 35, -30, 20, -5, 1, 31, -60, 105, -60, 30, -6, 1, -21, 217, -210, 245, -105, 42, -7, 1, 204, -168, 868, -560, 490, -168, 56, -8, 1, 307, 1836, -756, 2604, -1260, 882, -252, 72, -9, 1, 2811, 3070, 9180, -2520, 6510, -2520, 1470, -360, 90, -10, 1
Offset: 0

Views

Author

Paul Barry, Apr 23 2009

Keywords

Comments

First column is A126617. Row sums are A000296. A007318*A159830 is A124323.
The inverse is [exp(-exp(x)+1+2x),x] which has production matrix given by
1, 1,
-1, 1, 1,
-1, -2, 1, 1,
-1, -3, -3, 1, 1,
-1, -4, -6, -4, 1, 1 ...

Examples

			Triangle begins
1,
-1, 1,
2, -2, 1,
-3, 6, -3, 1,
7, -12, 12, -4, 1,
-10, 35, -30, 20, -5, 1,
31, -60, 105, -60, 30, -6, 1,
-21, 217, -210, 245, -105, 42, -7, 1,
204, -168, 868, -560, 490, -168, 56, -8, 1
Production array is
-1, 1,
1, -1, 1,
1, 2, -1, 1,
1, 3, 3, -1, 1,
1, 4, 6, 4, -1, 1,
1, 5, 10, 10, 5, -1, 1,
1, 6, 15, 20, 15, 6, -1, 1,
1, 7, 21, 35, 35, 21, 7, -1, 1,
1, 8, 28, 56, 70, 56, 28, 8, -1, 1

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Exp[#] - 1 - 2 #]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

G.f.: 1/(1-xy+x-x^2/(1-xy-2x^2/(1-xy-x-3x^2/(1-xy-2x-4x^2/(1-... (continued fraction).

A367938 Expansion of e.g.f. exp(exp(3*x) - 1 - 2*x).

Original entry on oeis.org

1, 1, 10, 55, 487, 4654, 51463, 632125, 8536492, 125279785, 1981246555, 33530245984, 603797462677, 11513675558701, 231539488842610, 4893151984630579, 108334206855000739, 2505977899186557502, 60419653270442268643, 1515077412621445514089, 39437350309301393464876, 1063746973172416765272589
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A126617, A247452, A284864, A367785, A367888.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] (-2)^(n - k) 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - 3*x)) / (1 - 3*x) ).
a(n) = exp(-1) * Sum_{k>=0} (3*k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * Bell(k).

A367939 Expansion of e.g.f. exp(exp(4*x) - 1 - 2*x).

Original entry on oeis.org

1, 2, 20, 168, 1936, 25376, 378688, 6284928, 114471168, 2263605760, 48192279552, 1097180784640, 26562251100160, 680591327567872, 18381995707154432, 521521320660205568, 15495495061984051200, 480873815489757970432, 15549555768325162926080, 522810678067316117733376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A126617, A355162, A367786, A367938, A367940.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-2)^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - 4*x)) / (1 - 4*x) ).
a(n) = exp(-1) * Sum_{k>=0} (4*k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n+k) * Bell(k).
a(n) = 2^n * |A124311(n)|.
Previous Showing 11-14 of 14 results.

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