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 A136580 Row sums of triangle A136579. 5
 1, 1, 3, 7, 27, 127, 747, 5167, 41067, 368047, 3669867, 40284847, 482671467, 6267305647, 87660962667, 1313941673647, 21010450850667, 357001369769647, 6423384156578667, 122002101778601647, 2439325392333218667 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Eric Weisstein's MathWorld, Incomplete Gamma Function. Eric Weisstein's MathWorld, Exponential Integral. FORMULA 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. 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 EXAMPLE 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 + ... MAPLE A128174 := proc(n, k) if k <= 0 or k >n then 0; else 1- ( (n-k) mod 2) ; end if; end proc: A136572 := proc(n, k) if k <> n then 0; else n! ; end if; end proc: A136579 := proc(n, k) add( A128174(n+1, j+1)*A136572(j, k), j=0..n) ; end proc: A136580 := proc(n) add( A136579(n, k), k=0..n) ; end proc: seq(A136580(n, k), n=0..60); # R. J. Mathar, Feb 22 2010 MATHEMATICA 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 *) CROSSREFS Cf. A136579, A000142, A153229, A058006. Sequence in context: A062363 A333362 A242328 * A292897 A267625 A161370 Adjacent sequences:  A136577 A136578 A136579 * A136581 A136582 A136583 KEYWORD nonn AUTHOR Gary W. Adamson, Jan 09 2008 EXTENSIONS More terms from R. J. Mathar, Feb 22 2010 STATUS approved

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Last modified April 20 15:08 EDT 2021. Contains 343135 sequences. (Running on oeis4.)