A166353 -id:A166353 - cp's OEIS frontend

A166354 Row sums of exponential Riordan array [1+x*tan(x/2),x], A166353.

1, 1, 2, 4, 8, 16, 34, 78, 200, 568, 1806, 6282, 24052, 99100, 443178, 2107966, 10775664, 58092112, 334087750, 2012990930, 12863046636, 85662585604, 602124105122, 4391793687974, 33676375206568, 266989039507576

Offset: 0

Paul Barry, Oct 12 2009

Binomial transform of aerated Genocchi number variant with e.g.f. 1+x*tan(x/2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A110501.

CoefficientList[Series[E^x*(1+x*Tan[x/2]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

E.g.f.: exp(x)*(1+x*tan(x/2)).
a(n)=sum{k=0..n, C(n,k)*G(k/2)(1+(-1)^k)/2} where
G(n)=0^n+2(-1)^n*(1-4^n)*sum{k=0..2n, sum{j=0..k, (-1)^j*C(k,j)*j^(2n)/(k+1)}}.
a(n) ~ n! * 2*(exp(Pi)+(-1)^n*exp(-Pi))/Pi^n. - Vaclav Kotesovec, Oct 02 2013

A166355 Diagonal sums of exponential Riordan array [1+x*tan(x/2),x], A166353.

Original entry on oeis.org

1, 2, 5, 15, 64, 443, 4887, 78996, 1745995, 50333929, 1829758158, 81753825477, 4399497764477, 280491321580150, 20898005984605281, 1798558057748753171, 177034863818072607020, 19758697171102806823327

Offset: 0

Paul Barry, Oct 12 2009

Aerated sequence gives diagonal sums of A166353.

Cf. A110501, A166353.

(* The function RiordanArray is defined in A256893. *)
nmax = 17; R = RiordanArray[1 + # Tan[#/2]&, #&, 2 nmax + 1, True];
a[n_] := Sum[R[[i, 2 n - i + 2]], {i, 2 n + 1, n + 1, -1}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 20 2019 *)

a(n)=sum{k=0..n, C(n+k,2k)*G(k)} where G(n)=0^n+2(-1)^n*(1-4^n)*sum{k=0..2n, sum{j=0..k, (-1)^j*C(k,j)*j^(2n)/(k+1)}}.

cp's OEIS Frontend

A166354 Row sums of exponential Riordan array [1+x*tan(x/2),x], A166353.

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