The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A050946 "Stirling-Bernoulli transform" of Fibonacci numbers. 7
 0, 1, 1, 7, 13, 151, 421, 6847, 25453, 532231, 2473141, 63206287, 352444093, 10645162711, 69251478661, 2413453999327, 17943523153933, 708721089607591, 5927841361456981, 261679010699505967, 2431910546406522973, 118654880542567722871, 1212989379862721528101 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS From Paul Curtz, Oct 11 2013: (Start) Differences table:      0,    1,    1,    7,   13,  151,  421, 6847, ...      1,    0,    6,    6,  138,  270, 6426, ...     -1,    6,    0,  132,  132, 6156, ...      7,   -6,  132,    0, 6024, ...    -13,  138, -132, 6024, ...    151, -270, 6156, ...   -421, 6426, ...   6847, ... . a(n) is an autosequence of first kind: the inverse binomial transform is the sequence signed, the main diagonal is A000004=0's. The "Stirling-Bernoulli transform" applied to an autosequence of first kind is an autosequence of first kind. Now consider the Akiyama-Tanigawa transform or algorithm applied to A000045(n):      0,   1,   1,   2,   3,   5, 8, ...     -1,   0,  -3,  -4, -10, -18, ...    = -A006490     -1,   6,   3,  24,  40, ...     -7,   6, -63, -64, ...    -13, 138,   3, ...   -151, 270, ...   -421, ... . Hence -a(n). The Akiyama-Tanigawa algorithm applied to an autosequence of first kind is an autosequence of first kind. a(n+5) - a(n+1) = 150, 420, 6840, ... is divisible by 30. 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. 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). 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) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..447 C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7 FORMULA O.g.f.: Sum_{n>=1} Fibonacci(n) * n! * x^n / Product_{k=1..n} (1+k*x). - Paul D. Hanna, Jul 20 2011 A100872(n)=a(2*n) and A100868(n)=a(2*n-1). From Paul Barry, Apr 20 2005: (Start) E.g.f.: exp(x)*(1-exp(x))/(1-3*exp(x)+exp((2*x))). a(n) = Sum_{k=0..n} (-1)^(n-k)*S2(n, k)*k!*Fibonacci(k). [corrected by Ilya Gutkovskiy, Apr 04 2019] (End) 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 a(n) = -1 * Sum_{k = 0..n} A163626(n,k)*A000045(k). - Philippe Deléham, May 29 2015 MAPLE with(combinat): a:= n-> add((-1)^(k+1) *k! *stirling2(n+1, k+1)*fibonacci(k), k=0..n): seq(a(n), n=0..30);  # Alois P. Heinz, May 17 2013 MATHEMATICA 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[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 *) PROG (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 */ CROSSREFS Cf. A000045, A051782, A105796, A163626. Sequence in context: A241280 A219703 A198947 * A178956 A319612 A247946 Adjacent sequences:  A050943 A050944 A050945 * A050947 A050948 A050949 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 02 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 3 02:56 EST 2020. Contains 338899 sequences. (Running on oeis4.)