A136579 -id:A136579 - cp's OEIS frontend

A136580 Row sums of triangle A136579.

1, 1, 3, 7, 27, 127, 747, 5167, 41067, 368047, 3669867, 40284847, 482671467, 6267305647, 87660962667, 1313941673647, 21010450850667, 357001369769647, 6423384156578667, 122002101778601647, 2439325392333218667

Offset: 0

Gary W. Adamson, Jan 09 2008

			a(4) = 27 = sum of row 4 terms, triangle A136579: (1 + 0 + 2 + 0 + 24) = 0! + 2! + 4!.
a(5) = 127 = sum of row 5 terms, triangle A136579: (0 + 1 + 0 + 6 + 0 + 120) = 1! + 3! + 5!
G.f. = 1 + x + 3*x^2 + 7*x^3 + 27*x^4 + 127*x^5 + 747*x^6 + 5167*x^7 + 41067*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..449
Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A136579, A000142, A153229, A058006.

Cf. A358498, A358499, A358500.

A136580 := proc(n)
    add( (n-2*i)!,i=0..floor(n/2) ) ;
end proc: # R. J. Mathar, Jun 04 2021

a[0] = 1; a[1] = 1; a[2] = 3; a[n_] := a[n] = n a[n-1] + a[n-2] - n a[n-3]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

G.f.: 2/(1-x^2)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: Q(0)/(1-x^2), where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
From Vladimir Reshetnikov, Oct 29 2015: (Start):
a(n) = (-1)^n*exp(1)*Gamma(0, 1)/2 - Re(Gamma(0, -1))*exp(-1)/2 + (n+2)!*((-1)^n*Re(Gamma(-n-2, -1))*exp(-1)-Gamma(-n-2, 1)*exp(1))/2, where Gamma(a, x) is the upper incomplete Gamma function.
D-finite with recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = n*a(n-1) + a(n-2) - n*a(n-3).
E.g.f.: 1/(1-x) + (exp(x-1)*(Ei(1)-Ei(1-x)) + exp(1-x)*(Ei(x-1)-Ei(-1)))/2, where Ei(x) is the exponential integral.
a(n+1)-a(n) = A153229(n+2) = (-1)^(n+1)*A058006(n+1).
(End)
0 = a(n)*(+a(n+1) - a(n+2) - a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - 2*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0. - Michael Somos, Oct 29 2015

A137948 Triangle read by rows, A000012 * A136579.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 6, 5, 4, 6, 12, 24, 6, 5, 8, 18, 48, 120, 7, 6, 10, 24, 72, 240, 720, 8, 7, 12, 30, 96, 360, 1440, 5040, 9, 8, 14, 36, 120, 480, 2160, 10080, 40320, 10, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880

Offset: 0

Gary W. Adamson, Feb 28 2008

Row sums = A014144 starting (1, 3, 7, 17, 51, 205, ...).

T(n,k) = A245334(n,k) / A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Aug 31 2014

			First few rows of the triangle:
  1;
  2, 1;
  3, 2,  2;
  4, 3,  4,  6;
  5, 4,  6, 12, 24;
  6, 5,  8, 18, 48, 120;
  7, 6, 10, 24, 72, 240, 720;
  ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A136579, A014144.

Cf. A245334, A007318.

a137948 n k = a137948_tabl !! n !! k
a137948_row n = a137948_tabl !! n
a137948_tabl = zipWith (zipWith div) a245334_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 31 2014

As infinite lower triangular matrices, A000012 * A136579, where A000012 = (1; 1,1; 1,1,1; ...) and A136579 = (1; 1,1; 1,1,2; 1,1,2,6; 1,1,2,6,24; ...).

T(n,k) = (n+1-k) * k! for 0 <= k <= n. - Werner Schulte, Oct 06 2020

Offset changed by Reinhard Zumkeller, Aug 31 2014

A138015 Triangle read by rows, antidiagonals of an array formed by A000012 * A136579. Replace the term "n" in the correlation triangle A003983 with A003422(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 10, 4, 2, 1, 1, 2, 4, 10, 10, 4, 2, 1, 1, 2, 4, 10, 34, 10, 4, 2, 1, 1, 2, 4, 10, 34, 34, 10, 4, 2, 1, 1, 2, 4, 10, 34, 154, 34, 10, 4, 2, 1, 1, 2, 4, 10, 34, 154, 154, 34, 10, 4, 2, 1

Offset: 1

Gary W. Adamson, Feb 28 2008

A003422 = (0, 1, 2, 4, 10, 34, 154, ...).

Row sums = A138016: (1, 2, 4, 6, 10, 14, 24, 34, 68, 102, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2,  1;
  1, 2, 4,  2,  1;
  1, 2, 4,  4,  2,  1;
  1, 2, 4, 10,  4,  2,  1;
  1, 2, 4, 10, 10,  4,  2, 1;
  1, 2, 4, 10, 34, 10,  4, 2, 1;
  1, 2, 4, 10, 34, 34, 10, 4, 2, 1;
  ...

Cf. A136579, A138016, A003983, A003422.

Example corrected by Philippe Deléham, Feb 18 2013

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A136580 Row sums of triangle A136579.

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A137948 Triangle read by rows, A000012 * A136579.

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A138015 Triangle read by rows, antidiagonals of an array formed by A000012 * A136579. Replace the term "n" in the correlation triangle A003983 with A003422(n).

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