cp's OEIS Frontend

A385641 Partial sums of A097893.

%I A385641 #19 Aug 28 2025 19:37:38
%S A385641 1,3,8,20,51,133,356,972,2695,7557,21372,60840,174097,500295,1442720,
%T A385641 4172752,12099411,35161001,102375400,298586652,872177273,2551118623,
%U A385641 7471195500,21904500500,64286141881,188844619563,555216323396,1633658183432,4810340397375,14173698242137
%N A385641 Partial sums of A097893.
%C A385641 Second partial sums of the central trinomial coefficients (A002426).
%C A385641 Third partial sums of A025178 (sequence starting 1, 0, 2, 4, 12, 32, 90 .... with offset 0).
%C A385641 For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).
%C A385641 For p Pythagorean prime (A002144), a(p) - 3 == 0 (mod p).
%C A385641 Sequences with g.f. (1-x)^k / sqrt(1-2*x-3*x^2): this sequence (k=-2), A097893 (k=-1), A002426 (k=0), A025178 (k=1), A024997 (k=2), A026083 (k=3). - _Mélika Tebni_, Aug 25 2025
%F A385641 G.f.: (1 / sqrt((1 + x)*(1 - 3*x))) / (1 - x)^2.
%F A385641 E.g.f.: exp(x)*(BesselI(0, 2*x) + 2*g(x) + Integral_{x=-oo..oo} g(x) dx) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx.
%F A385641 D-finite with recurrence n*a(n) = (4*n-1)*a(n-1) - (2*n+1)*a(n-2) - (4*n-5)*a(n-3) + 3*(n-1)*a(n-4).
%F A385641 a(0) = 1, a(1) = 3 and a(n) = a(n-2) - 1 + 2*A383527(n) for n >= 2.
%F A385641 a(n) = Sum_{k=0..n} binomial(n+1, k+1)*A128014(k).
%F A385641 a(n) = Sum_{k=0..n} (2*A247287(k) + k+1).
%F A385641 a(n) ~ 3^(n + 5/2) / (8*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 03 2025
%p A385641 a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);
%o A385641 (Python)
%o A385641 from math import comb as C
%o A385641 def a(n):
%o A385641     return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
%o A385641 print([a(n) for n in range(30)])
%o A385641 (PARI) a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ _Michel Marcus_, Aug 06 2025
%Y A385641 Cf. A002144, A002145, A002426, A024997, A025178, A026083,  A097893, A128014, A247287, A383527.
%K A385641 nonn,easy,changed
%O A385641 0,2
%A A385641 _Mélika Tebni_, Aug 03 2025