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A145561 Alternating row sums of triangle A049029 (S2(5)).

%I A145561 #18 Jan 20 2025 09:02:33
%S A145561 1,4,31,359,5546,107249,2492701,67693534,2103854581,73651161959,
%T A145561 2868077514776,122980857764819,5758029769553101,292305762924889804,
%U A145561 15992593021331060611,938143525674896325299,58739433900424758545186,3910020681156059085488189
%N A145561 Alternating row sums of triangle A049029 (S2(5)).
%C A145561 a(537) < 0. - _Seiichi Manyama_, Jan 20 2025
%F A145561 a(n) = Sum_{m=1..n} (-1)^(m+1)*A049029(n,m), n>=1.
%F A145561 E.g.f.: (from Jabotinsky structure): 1-exp(1-1/(1-4*x)^(1/4)).
%F A145561 a(n) = y(n), where y(0) = -1, y(1) = 1, y(2) = 4, y(3) = 31, y(4) = 359, and -32*k*(1 + k)*(1 + 2 k)*(1 + 4 k)*(3 + 4 k)*y(k) + (1679 + 5920 k + 8080 k^2 + 5120 k^3 + 1280 k^4)*y(k+1) + (-2550 - 4580 k - 2880 k^2 - 640 k^3)*y(k+2) + (675 + 640 k + 160 k^2)*y(k+3) + (-50 - 20 k)*y(k+4) + y(k+5) = 0. - _Benedict W. J. Irwin_, Jul 12 2017
%F A145561 From _Seiichi Manyama_, Jan 20 2025: (Start)
%F A145561 a(n) = -Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A000587(k).
%F A145561 a(n) = -e * (-4)^n * n! * Sum_{k>=0} (-1)^k * binomial(-k/4,n)/k!. (End)
%t A145561 Table[DifferenceRoot[Function[{y, k}, {-32 k (1 + k) (1 + 2 k) (1 + 4 k) (3 + 4 k) y[k] + (1679 + 5920 k + 8080 k^2 + 5120 k^3 + 1280 k^4) y[1 + k] + (-2550 - 4580 k - 2880 k^2 - 640 k^3) y[2 + k] + (675 + 640 k + 160 k^2) y[3 + k] + (-50 - 20 k) y[4 + k] + y[5 + k] == 0, y[0] == -1, y[1] == 1, y[2] == 4, y[3] == 31, y[4] == 359}]][n], {n, 1, 20}] (* _Benedict W. J. Irwin_, Jul 12 2017 *)
%Y A145561 Cf. A049029, A049120 (row sums).
%K A145561 sign,easy
%O A145561 1,2
%A A145561 _Wolfdieter Lang_, Oct 17 2008