A093951 -id:A093951 - cp's OEIS frontend

A167763 Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 7, 4, 1, 0, 1, 5, 12, 12, 5, 1, 0, 1, 6, 18, 30, 18, 6, 1, 0, 1, 7, 25, 55, 55, 25, 7, 1, 0, 1, 8, 33, 88, 143, 88, 33, 8, 1, 0, 1, 9, 42, 130, 273, 273, 130, 42, 9, 1, 0, 1, 10, 52, 182, 455, 728, 455, 182, 52, 10, 1, 0

Offset: 0

Philippe Deléham, Nov 11 2009

See A118340 for definition of pendular triangles and pendular sums.

The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - Tom Copeland, Jan 22 2020

			Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,  0;
  1,  3,  3,  1,  0;
  1,  4,  7,  4,  1,  0;
  1,  5, 12, 12,  5,  1,  0; ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).

Cf. A001764, A006013, A006629, A093951, A102893, A173075.

function T(n,k,p)
  if k lt 0 or n lt k then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);
  else return T(n,n-k-1,p) + p*T(n-1,k,p);
  end if;
  return T;
end function;
[T(n,k,0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_, p_]:= T[n,k,p] = If[nG. C. Greubel, Feb 17 2021 *)

{T(n,k)=if(k==0,1,if(n==k,0,if(n>2*k,binomial(n+k+1,k)*(n-2*k+1)/(n+k+1),T(n,n-1-k))))} \\ Paul D. Hanna, Nov 12 2009

@CachedFunction
def T(n, k, p):
    if (k<0 or n2*k): return T(n,n-k,p) + T(n-1,k,p)
    else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.
If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - Paul D. Hanna, Nov 12 2009
Sum_{k=0..n} T(n, k, p=0) = A093951(n). - G. C. Greubel, Feb 17 2021

A084081 Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.

Original entry on oeis.org

0, 1, 2, 5, 10, 24, 50, 121, 260, 637, 1400, 3468, 7752, 19380, 43890, 110561, 253000, 641355, 1480050, 3771885, 8765250, 22439040, 52451256, 134796060, 316663760, 816540124, 1926501200, 4982228488, 11798983280, 30593078076, 72690164850

Offset: 0

Wouter Meeussen, May 11 2003

nonn

Lengths of lists is A047749.

			Lists {0}, {1}, {2, 0}, {3, 1, 1}, {4, 2, 0, 2, 0, 2, 0} sum to 0, 1, 2, 5, 10.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A047749, A093951.

F:=Floor; B:=Binomial;
function A084081(n)
  if (n mod 2) eq 0 then return 10*B(F((3*n+2)/2), F((n-2)/2))/(n+3);
  else return 2*(3*n+1)*B(F((3*n+5)/2), F((n+1)/2))/((n+3)*(3*n+5));
  end if; return A084081;
end function;
[A084081(n): n in [0..40]]; // G. C. Greubel, Oct 17 2022

Plus@@@Flatten/@NestList[ # /. k_Integer :> Range[k+1, 0, -2]&, {0}, 8]
A084081[n_]:= If[EvenQ[n], 10*Binomial[(3*n+2)/2, (n-2)/2]/(n+3), 2*(3*n + 1)*Binomial[(3*n+5)/2, (n+1)/2]/((n+3)*(3*n+5))];
Table[A084081[n], {n, 40}] (* G. C. Greubel, Oct 17 2022 *)

def A084081(n):
    if (n%2==0): return 10*binomial(int((3*n+2)/2), int((n-2)/2))/(n+3)
    else: return 2*(3*n+1)*binomial(int((3*n+5)/2), int((n+1)/2))/((n+3)*(3*n+5))
[A084081(n) for n in range(40)] # G. C. Greubel, Oct 17 2022

Equals A093951(n) - A047749(n).
From G. C. Greubel, Oct 17 2022: (Start)
a(2*n+1) = (3*n-1)*binomial[3*n+1, n]/((n+1)*(3*n+1)).
a(2*n) = 10*binomial(3*n+1, n-1)/(2*n+3). (End)

cp's OEIS Frontend

A167763 Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula