This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A257838 #37 Apr 09 2017 02:58:29
%S A257838 0,1,4,16,63,247,967,3785,14820,58060,227612,892926,3505386,13770404,
%T A257838 54129602,212904952,837885495,3299264407,12997784803,51230474669,
%U A257838 202014314769,796928589755,3145066003589,12416625685891,49037912997003,193734379979677,765632076098287,3026670770970925,11968378998073935
%N A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).
%C A257838 The array used here starts in row n=0 with the first partial sums of A000045. The array which starts with the Fibonacci numbers in row k=0 is shown in A136431. The diagonal of that array is given in A176085. - _Wolfdieter Lang_, Jun 03 2015
%H A257838 G. C. Greubel, <a href="/A257838/b257838.txt">Table of n, a(n) for n = 0..1000</a>
%F A257838 a(n) = F^{n+1}(n), n >= 0, with the k-th iterated partial sum F^{k} of the Fibonacci number A000045. - _Wolfdieter Lang_, Jun 03 2015
%F A257838 Conjecture: n*(n-3)*a(n) +2*(-4*n^2+13*n-6)*a(n-1) +(15*n^2-53*n+48)*a(n-2) +2*(2*n-3)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Dec 10 2015
%F A257838 G.f.: -(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x). - _Vladimir Kruchinin_, Oct 09 2016
%F A257838 a(n) = Sum_{k=0..n} binomial(2*n-k,n-k)*F(k), where F(k) = A000045(k). - _Vladimir Kruchinin_, Oct 09 2016
%F A257838 a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 09 2016
%e A257838 This sequence is the main diagonal of the following array (see the comment and Example field of A136431):
%e A257838 0, 1, 2, 4, 7, 12, ... A000071
%e A257838 0, 1, 3, 7, 14, 26, ... A001924
%e A257838 0, 1, 4, 11, 25, 51, ... A014162
%e A257838 0, 1, 5, 16, 41, 92, ... A014166
%e A257838 0, 1, 6, 22, 63, 155, ... A053739
%e A257838 0, 1, 7, 29, 92, 247, ... A053295
%t A257838 Table[DifferenceRoot[Function[{a, n},{(2*n + 4*n^2)*a[n] + (2 + 7*n + 15*n^2)*a[1 + n] + (8 - 6*n - 8*n^2)*a[2 + n] + (-2 + n + n^2)*a[3 + n] == 0, a[1] == 0, a[2] == 1, a[3] == 4, a[4] == 16}]][n], {n, 30}]
%o A257838 (Maxima)
%o A257838 a(n):=sum(binomial(2*n-k,n-k)*fib(k),k,0,n); /* _Vladimir Kruchinin_, Oct 09 2016 */
%o A257838 (PARI) x='x+O('x^50); concat([0], Vec(-(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x))) \\ _G. C. Greubel_, Apr 08 2017
%Y A257838 Cf. A000045, A000071, A001924, A014162, A014166, A136431, A176085.
%K A257838 nonn,easy
%O A257838 0,3
%A A257838 _Luciano Ancora_, May 10 2015
%E A257838 Name edited by _Wolfdieter Lang_, Jun 03 2015