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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A292632 a(n) = n! * [x^n] exp((n+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)).

Original entry on oeis.org

1, 2, 10, 77, 798, 10392, 162996, 2991340, 62893270, 1490758022, 39334017996, 1143492521437, 36318168041260, 1251270023475864, 46481870133666792, 1852054390616046345, 78792796381529620710, 3564894013016856836190, 170921756533520140861020, 8657018996674423681277455, 461881087606113071895396420
Offset: 0

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Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A000108.

Links

Crossrefs

Main diagonal of A271025.
Cf. A000108, A292631, A349603.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[E^((n+2)*x)*(BesselI[0,2*x] - BesselI[1,2*x]),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)
Join[{1}, Table[Sum[Binomial[n, j] * CatalanNumber[j] * n^(n-j), {j, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 23 2021 *)

Formula

a(n) = [x^n] (sqrt(1 - n*x) - sqrt(1 - 4*x - n*x))/(2*x*sqrt(1 - n*x)).
a(n) = A271025(n,n).
a(n) ~ exp(2) * (BesselI(0,2) - BesselI(1,2)) * n^n. - Vaclav Kotesovec, Sep 20 2017
a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * n^(n-k). - Vaclav Kotesovec, Nov 23 2021

A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 7, 27, 1, 4, 12, 43, 256, 1, 5, 19, 71, 393, 3125, 1, 6, 28, 117, 616, 4721, 46656, 1, 7, 39, 187, 985, 7197, 69853, 823543, 1, 8, 52, 287, 1584, 11123, 105052, 1225757, 16777216, 1, 9, 67, 423, 2521, 17429, 159093, 1829291, 24866481, 387420489, 1, 10, 84, 601, 3928, 27525, 243256, 2740111, 36922928, 572410513, 10000000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A000312 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 4)*x^2/2! + (k^3 + 3*k^2 + 12*k + 27)*x^3/3! + (k^4 + 4*k^3 + 24*k^2 + 108*k + 256)*x^4/4! + ...
Square array begins:
     1,     1,     1,     1,     1,     1, ...
     1,     2,     3,     4,     5,     6, ...
     4,     7,    12,    19,    28,    39, ...
    27,    43,    71,   117,   187,   287, ...
   256,   393,   616,   985,  1584,  2521, ...
  3125,  4721,  7197, 11123, 17429, 27525, ...

Links

Crossrefs

Columns k=0..2 give A000312, A086331, A277457.
Main diagonal gives A290840.
Cf. A080955, A081576, A271025.

Programs

  • Mathematica
    Table[Function[k, n!*SeriesCoefficient[Exp[k x]/(1 + LambertW[-x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten (* G. C. Greubel, Nov 09 2017 *)

Formula

E.g.f. of column k: exp(k*x)/(1 + LambertW(-x)).
A(n,k) = Sum_{j=0..n} binomial(n,j)*k^(n-j)*j^j. - Fabian Pereyra, Jul 16 2024
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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