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A050946 "Stirling-Bernoulli transform" of Fibonacci numbers.

%I A050946 #55 Apr 05 2019 09:26:42
%S A050946 0,1,1,7,13,151,421,6847,25453,532231,2473141,63206287,352444093,
%T A050946 10645162711,69251478661,2413453999327,17943523153933,708721089607591,
%U A050946 5927841361456981,261679010699505967,2431910546406522973,118654880542567722871,1212989379862721528101
%N A050946 "Stirling-Bernoulli transform" of Fibonacci numbers.
%C A050946 From _Paul Curtz_, Oct 11 2013: (Start)
%C A050946 Differences table:
%C A050946      0,    1,    1,    7,   13,  151,  421, 6847, ...
%C A050946      1,    0,    6,    6,  138,  270, 6426, ...
%C A050946     -1,    6,    0,  132,  132, 6156, ...
%C A050946      7,   -6,  132,    0, 6024, ...
%C A050946    -13,  138, -132, 6024, ...
%C A050946    151, -270, 6156, ...
%C A050946   -421, 6426, ...
%C A050946   6847, ... .
%C A050946 a(n) is an autosequence of first kind: the inverse binomial transform is the sequence signed, the main diagonal is A000004=0's.
%C A050946 The "Stirling-Bernoulli transform" applied to an autosequence of first kind is an autosequence of first kind.
%C A050946 Now consider the Akiyama-Tanigawa transform or algorithm applied to A000045(n):
%C A050946      0,   1,   1,   2,   3,   5, 8, ...
%C A050946     -1,   0,  -3,  -4, -10, -18, ...    = -A006490
%C A050946     -1,   6,   3,  24,  40, ...
%C A050946     -7,   6, -63, -64, ...
%C A050946    -13, 138,   3, ...
%C A050946   -151, 270, ...
%C A050946   -421, ... .
%C A050946 Hence -a(n). The Akiyama-Tanigawa algorithm applied to an autosequence of first kind is an autosequence of first kind.
%C A050946 a(n+5) - a(n+1) = 150, 420, 6840, ... is divisible by 30.
%C A050946 For an autosequence of the second kind, the inverse binomial transform is the sequence signed with the main diagonal double of the first upper diagonal.
%C A050946 The Akiyama-Tanigawa algorithm applied to an autosequence leads to an autosequence of the same kind. Example: the A-T algorithm applied to the autosequence of second kind 1/n leads to the autosequence of the second kind A164555(n)/A027642(n).
%C A050946 Note that a2(n) = 2*a1(n+1) - a1(n) applied to the autosequence of the first kind a1(n) is a corresponding autosequence of the second kind. (End)
%H A050946 Alois P. Heinz, <a href="/A050946/b050946.txt">Table of n, a(n) for n = 0..447</a>
%H A050946 C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7
%F A050946 O.g.f.: Sum_{n>=1} Fibonacci(n) * n! * x^n / Product_{k=1..n} (1+k*x). - _Paul D. Hanna_, Jul 20 2011
%F A050946 A100872(n)=a(2*n) and A100868(n)=a(2*n-1).
%F A050946 From _Paul Barry_, Apr 20 2005: (Start)
%F A050946 E.g.f.: exp(x)*(1-exp(x))/(1-3*exp(x)+exp((2*x))).
%F A050946 a(n) = Sum_{k=0..n} (-1)^(n-k)*S2(n, k)*k!*Fibonacci(k). [corrected by _Ilya Gutkovskiy_, Apr 04 2019] (End)
%F A050946 a(n) ~ c * n! / (log((3+sqrt(5))/2))^(n+1), where c = 1/sqrt(5) if n is even and c = 1 if n is odd. - _Vaclav Kotesovec_, Aug 13 2013
%F A050946 a(n) = -1 * Sum_{k = 0..n} A163626(n,k)*A000045(k). - _Philippe Deléham_, May 29 2015
%p A050946 with(combinat):
%p A050946 a:= n-> add((-1)^(k+1) *k! *stirling2(n+1, k+1)*fibonacci(k), k=0..n):
%p A050946 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 17 2013
%t A050946 CoefficientList[Series[E^x*(1-E^x)/(1-3*E^x+E^(2*x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Aug 13 2013 *)
%t A050946 t[0, k_] := Fibonacci[k]; t[n_, k_] := t[n, k] = (k+1)*(t[n-1, k] - t[n-1, k+1]); a[n_] := t[n, 0] // Abs; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Oct 22 2013, after _Paul Curtz_ *)
%o A050946 (PARI) {a(n)=polcoeff(sum(m=0, n, fibonacci(m)*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* _Paul D. Hanna_, Jul 20 2011 */
%Y A050946 Cf. A000045, A051782, A105796, A163626.
%K A050946 nonn
%O A050946 0,4
%A A050946 _N. J. A. Sloane_, Jan 02 2000