login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A154231 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1), read by rows. 6
1, 1, 1, 1, 278, 1, 1, 1579, 1579, 1, 1, 6005, 1233308, 6005, 1, 1, 18207, 20504692, 20504692, 18207, 1, 1, 47216, 194715939, 35816807848, 194715939, 47216, 1, 1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1, 1, 229819, 7024500980, 24830582225241, 4330171226988158, 24830582225241, 7024500980, 229819, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 280, 3160, 1245320, 41045800, 36206334160, ...}.

The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12 = A000539(n+1). - G. C. Greubel, Mar 02 2021

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

EXAMPLE

Triangle begins as:

  1;

  1,      1;

  1,    278,          1;

  1,   1579,       1579,             1;

  1,   6005,    1233308,          6005,             1;

  1,  18207,   20504692,      20504692,         18207,          1;

  1,  47216,  194715939,   35816807848,     194715939,      47216,      1;

  1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1;

MAPLE

T:= proc(n, k) option remember;

      if k=0 or k=n then 1

    else T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T(n-2, k-1)

      fi; end:

seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021

MATHEMATICA

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T[n-2, k-1] ];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

PROG

(Sage)

def f(n): return binomial(n+2, 2)^2*(2*n^2+6*n+3)/3

def T(n, k):

    if (k==0 or k==n): return 1

    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

(Magma)

f:= func< n | Binomial(n+2, 2)^2*(2*n^2+6*n+3)/3 >;

function T(n, k)

  if k eq 0 or k eq n then return 1;

  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);

  end if; return T;

end function;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

CROSSREFS

Cf. A154227, A154228, A154229, A154230, A154233.

Cf. A000539 (powers of 5).

Sequence in context: A048525 A189609 A264371 * A252249 A257368 A056995

Adjacent sequences:  A154228 A154229 A154230 * A154232 A154233 A154234

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Jan 05 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 02 2021

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 27 10:39 EST 2022. Contains 350607 sequences. (Running on oeis4.)