A001655 -id:A001655 - cp's OEIS frontend

A110112 Square array of numbers associated to the recurrences b(k) = b(k-1) + n*b(k-2); array T(n,k), read by descending antidiagonals, for n, k >= 0.

1, 1, 1, 1, 3, 1, 1, 15, 5, 1, 1, 60, 55, 7, 1, 1, 260, 385, 133, 9, 1, 1, 1092, 3311, 1330, 261, 11, 1, 1, 4641, 25585, 18430, 3393, 451, 13, 1, 1, 19635, 208335, 210490, 68237, 7216, 715, 15, 1, 1, 83215, 1652145, 2673223, 1037673, 197456, 13585, 1065, 17, 1, 1

Offset: 0

Paul Barry, Jul 12 2005

Rows include A001655, (-1)^n*A015266(n+3), A110111.

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1,  1,   1,    1,      1,       1,        1,          1, ...
  1,  3,  15,   60,    260,    1092,     4641,      19635, ...
  1,  5,  55,  385,   3311,   25585,   208335,    1652145, ...
  1,  7, 133, 1330,  18430,  210490,  2673223,   31940881, ...
  1,  9, 261, 3393,  68237, 1037673, 18598293,  300963537, ...
  1, 11, 451, 7216, 197456, 3761296, 89565861, 1842200151, ...
  ...

Cf. A083856.

a := proc(n, k) local v; option remember; if k = 0 and 0 <= n then v := 0; end if; if k = 1 and 0 <= n then v := 1; end if; if 2 <= k and 0 <= n then v := a(n, k - 1) + n*a(n, k - 2); end if; v; end proc;
T := proc(n, k) a(n, k + 1)*a(n, k + 2)*a(n, k + 3)/(n + 1); end proc;
seq(seq(T(k,n-k), k=0..n), n=0..10); # Petros Hadjicostas, Dec 26 2019

T(n, k) = a(n, k+1) * a(n, k+2) * a(n, k+3)/(n+1), where a(n, k) is the solution to a(n, k) = a(n, k-1) + n*a(n, k-2) for k >= 2 with a(n, 0) = 0 and a(n, 1) = 1 for all n >= 0.

Row n has g.f. 1/((1 + n*x - n^3*x^2) * (1 - (3*n + 1)*x - n^3*x^2)).

A177727 a(0)=1; a(n) = a(n-1) * Fibonacci(3+n) * Fibonacci(1+n) / (Fibonacci(n))^2, n > 1.

Original entry on oeis.org

1, 3, 30, 180, 1300, 8736, 60333, 412335, 2829310, 19384200, 132882696, 910735488, 6242420665, 42785803515, 293259265950, 2010026277756, 13776931957468, 94428478367520, 647222466507045, 4436128656563175, 30405678471399166, 208403619747957648, 1428419662108160400

Offset: 0

Roger L. Bagula, May 12 2010

Similar recurrences a(n) = a(n-1)*F(a0+n-1)*F(b0+n-1)/(F(n)*F(c0+n-1)) are:
{a0,b0,c0} = {3,2,1} in A066258.
{a0,b0,c0} = {3,1,1} in A001654.
{a0,b0,c0} = {4,1,1} in A001655 and next for 5,6 as well.

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, p. 93.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1)

Cf. A066258, A001654, A001655, A001656, A001657.

I:=[1, 3, 30, 180, 1300]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Nov 18 2011

with (combinat):
A177727 := proc(n)
   if n = 0 then
           1;
   else
           procname(n-1)*fibonacci(3+n)*fibonacci(1+n)/fibonacci(n)^2 ;
   end if;
end proc:
seq(A177727(n),n=0..10) ; # R. J. Mathar, Nov 17 2011

a0 = 4; b0 = 2; c0 = 1;
a[0] = 1;
a[n_] := a[n] = (Fibonacci[(a0 + n - 1)]*Fibonacci[( b0 + n - 1)]/(Fibonacci[n]*Fibonacci[(c0 + n - 1)]))*a[n - 1];
Table[a[n], {n, 0, 30}]
LinearRecurrence[{5,15,-15,-5,1},{1,3,30,180,1300},30] (* Vincenzo Librandi, Nov 18 2011 *)

G.f.: ( -1+2*x ) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - R. J. Mathar, Nov 17 2011

a(n) = A001656(n) - 2*A001656(n-1). - R. J. Mathar, Nov 17 2011

A215038 Partial sums of A066259: a(n) = Sum_{k=0..n} F(k+1)^2*F(k), n>=0, with the Fibonacci numbers F=A000045.

Original entry on oeis.org

0, 1, 5, 23, 98, 418, 1770, 7503, 31779, 134629, 570284, 2415788, 10233404, 43349461, 183631161, 777874251, 3295127934, 13958386366, 59128672790, 250473078515, 1061020985255, 4494557022121, 19039249069560, 80651553307128

Offset: 0

Wolfdieter Lang, Aug 09 2012

For a derivation of the explicit form of this sum see the link under A215308 on the partial summation formula, eq. (7).

			a(2) = 0 + 1^2*1 + 2^2*1 = 1 + 4 = 5.

Index entries for linear recurrences with constant coefficients, signature (4,3,-9,2,1).

Cf. A000045, A001655, A008346, A066258, A066259, A215308, A215037.

a(n) = Sum_{k=0..n} A066259(k) = Sum_{k=0..n} F(k+1)^2*F(k), n >= 0, with A066259(0)=0.
a(n) = (F(n+2)*F(n+1)^2 - (-1)^n*F(n) - 1)/2 = (A066258(n+1) - (-1)^n*A008346(n))/2, n >= 0.
O.g.f.: x*(1+x)/((1+x-x^2)*(1-4*x-x^2)*(1-x)) (from A066259).
E.g.f.: (2*exp(-x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2)) + exp(2*x)*(15*cosh(sqrt(15)*x) + 7*sqrt(5)*sinh(sqrt(5)*x)) - 25*exp(x))/50. - Stefano Spezia, Oct 28 2024

cp's OEIS Frontend

A110112 Square array of numbers associated to the recurrences b(k) = b(k-1) + n*b(k-2); array T(n,k), read by descending antidiagonals, for n, k >= 0.

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A177727 a(0)=1; a(n) = a(n-1) * Fibonacci(3+n) * Fibonacci(1+n) / (Fibonacci(n))^2, n > 1.

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A215038 Partial sums of A066259: a(n) = Sum_{k=0..n} F(k+1)^2*F(k), n>=0, with the Fibonacci numbers F=A000045.

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