A052179 -id:A052179 - cp's OEIS frontend

A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

1, 3, 1, 10, 6, 1, 37, 29, 9, 1, 150, 134, 57, 12, 1, 654, 622, 318, 94, 15, 1, 3012, 2948, 1686, 616, 140, 18, 1, 14445, 14317, 8781, 3693, 1055, 195, 21, 1, 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1, 361114, 360602, 238170, 118042, 44303, 12345, 2464, 332, 27, 1

Offset: 0

Philippe Deléham, Dec 11 2009

Equal to A171515*B = B*A104259, B = A007318.

			Triangle T(n,k) begins
[0]     1;
[1]     3,     1;
[2]    10,     6,     1;
[3]    37,    29,     9,     1;
[4]   150,   134,    57,    12,    1;
[5]   654,   622,   318,    94,   15,    1;
[6]  3012,  2948,  1686,   616,  140,   18,   1;
[7] 14445, 14317,  8781,  3693, 1055,  195,  21,  1;
[8] 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1;
.
Production array begins
  3, 1
  1, 3, 1
  1, 1, 3, 1
  1, 1, 1, 3, 1
  1, 1, 1, 1, 3, 1
  1, 1, 1, 1, 1, 3, 1
- _Philippe Deléham_, Mar 05 2013

Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -1, 0, 1, 2 respectively.

Cf. A007318, A052179, A171589, A171515, A104259.

T := proc(n,k) option remember;
if n < 0 or k < 0 then 0 elif n = k then 1 else
T(n-1, k-1) + 3*T(n-1,k) + add(T(n-1, k+1+i), i=0..n) fi end:
for n from 0 to 8 do seq(T(n,k), k = 0..n) od; # Peter Luschny, Oct 16 2022

T[n_, k_] := T[n, k] = If[n < 0 || k < 0, 0, If[n == k, 1, T[n-1, k-1] + 3*T[n-1, k] + Sum[T[n-1, k+1+i], {i, 0, n}]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 23 2024, after Peter Luschny *)

T(n, 0) - T(n, 1) = 2^n.

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + Sum_{i=0..n} T(n-1, k+1+i). - Philippe Deléham, Feb 23 2012

Corrected and extended by Peter Luschny, Oct 16 2022

A052178 Number of walks of length n on the simple cubic lattice terminating at height 2 above the (x,y)-plane.

Original entry on oeis.org

1, 12, 99, 700, 4569, 28476, 172508, 1026288, 6033690, 35195512, 204232809, 1181052756, 6814746393, 39267916380, 226097749224, 1301403695520, 7490649175326, 43123589230824, 248351880642630, 1430956006648056, 8249467230853002, 47587180659332248

Offset: 2

N. J. A. Sloane, Jan 26 2000

Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Column 2 of A052179.

b:= proc(n, k) option remember; `if`(min(n, k)<0, 0,
     `if`(max(n, k)=0, 1, b(n-1, k-1)+4*b(n-1, k)+b(n-1, k+1)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..25);  # Alois P. Heinz, Oct 28 2021

b[n_, k_] := b[n, k] = If[Min[n, k] < 0, 0, If[Max[n, k] == 0, 1, b[n - 1, k - 1] + 4*b[n - 1, k] + b[n - 1, k + 1]]];
a[n_] :=  b[n, 2];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jan 07 2025, after Alois P. Heinz *)

More terms and title improved by Sean A. Irvine, Oct 28 2021

A171515 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033543.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 16, 14, 6, 1, 62, 52, 27, 8, 1, 270, 213, 116, 44, 10, 1, 1257, 948, 513, 216, 65, 12, 1, 6096, 4470, 2376, 1038, 360, 90, 14, 1, 30398, 21904, 11468, 5056, 1880, 556, 119, 16, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to B^2*A039598*B^(-2), B = A007318.

			Triangle begins : 1 ; 2,1 : 5,4,1 ; 16,14,6,1 ; 62,52,27,8,1 ; ...

Cf. A052179, A171486

Sum_{k, 0<=k<=n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = 0, 1, 2, 3 respectively.

T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*2^i. - Philippe Deléham, Feb 23 2012

cp's OEIS Frontend

A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A052178 Number of walks of length n on the simple cubic lattice terminating at height 2 above the (x,y)-plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A171515 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033543.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula