A054341 -id:A054341 - cp's OEIS frontend

A202847 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126930.

1, -1, 1, 2, -2, 1, -3, 5, -3, 1, 6, -10, 9, -4, 1, -10, 22, -22, 14, -5, 1, 20, -44, 54, -40, 20, -6, 1, -35, 93, -123, 109, -65, 27, -7, 1, 70, -186, 281, -276, 195, -98, 35, -8, 1, -126, 386, -618, 682, -541, 321, -140, 44, -9, 1

Offset: 0

Philippe Deléham, Mar 02 2013

			Triangle begins
1
-1, 1
2, -2, 1
-3, 5, -3, 1
6, -10, 9, -4, 1
-10, 22, -22, 14, -5, 1
20, -44, 54, -40, 20, -6, 1
-35, 93, -123, 109, -65, 27, -7, 1
...
Production matrix begins
x, 1
1, x, 1
1, 1, x, 1
1, 1, 1, x, 1
1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, x, 1
..., with x = -1.

Cf. (sequences with similar production matrix) A097609 (x=0), A033184 (x=1), A104259 (x=2), A171568 (x=3), A171589 (x=4)

T(n,k) = (-1)^(n-k)*A054336(n,k).

Sum_{k, 0<=k<=n}T(n,k)*x^k = (-1)^n*A126931(n), (-1)^n*A054341(n), A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

A330019 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(k,floor(k/2)) * x^k / k!).

Original entry on oeis.org

1, 1, 4, 21, 150, 1330, 14180, 176295, 2505230, 40049226, 711379872, 13899553206, 296270826852, 6841305568812, 170127212242416, 4532854743105975, 128824523061126750, 3890041395675793930, 124375112406132404960, 4197530354920789582410, 149118181703716510545260

Offset: 0

Ilya Gutkovskiy, Nov 27 2019

nonn

Cf. A001405, A054341, A305561, A308849, A328004.

nmax = 20; CoefficientList[Series[1/(2 - BesselI[0, 2 x] - BesselI[1, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: 1 / (2 - BesselI(0,2*x) - BesselI(1,2*x)).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001405(k) * a(n-k).

cp's OEIS Frontend

A202847 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126930.

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