A036681 -id:A036681 - cp's OEIS frontend

A036355 Fibonacci-Pascal triangle read by rows.

1, 1, 1, 2, 2, 2, 3, 5, 5, 3, 5, 10, 14, 10, 5, 8, 20, 32, 32, 20, 8, 13, 38, 71, 84, 71, 38, 13, 21, 71, 149, 207, 207, 149, 71, 21, 34, 130, 304, 478, 556, 478, 304, 130, 34, 55, 235, 604, 1060, 1390, 1390, 1060, 604, 235, 55, 89, 420, 1177, 2272, 3310, 3736, 3310, 2272, 1177, 420, 89

Offset: 0

Floor van Lamoen, Dec 28 1998

T(n,k) is the number of lattice paths from (0,0) to (n-k,k) using steps (1,0),(2,0),(0,1),(0,2). - Joerg Arndt, Jun 30 2011, corrected by Greg Dresden, Aug 25 2020
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			Triangle begins
   1;
   1,   1;
   2,   2,   2;
   3,   5,   5,    3;
   5,  10,  14,   10,    5;
   8,  20,  32,   32,   20,    8;
  13,  38,  71,   84,   71,   38,   13;
  21,  71, 149,  207,  207,  149,   71,  21;
  34, 130, 304,  478,  556,  478,  304, 130,  34;
  55, 235, 604, 1060, 1390, 1390, 1060, 604, 235, 55;
with indices
  T(0,0);
  T(1,0),  T(1,1);
  T(2,0),  T(2,1),  T(2,2);
  T(3,0),  T(3,1),  T(3,2),  T(3,3);
  T(4,0),  T(4,1),  T(4,2),  T(4,3),  T(4,4);
For example, T(4,2) = 14 and there are 14 lattice paths from (0,0) to (4-2,2) = (2,2) using steps (1,0),(2,0),(0,1),(0,2). - _Greg Dresden_, Aug 25 2020

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Row sums form sequence A002605. T(n, 0) forms the Fibonacci sequence (A000045). T(n, 1) forms sequence A001629.
Derived sequences: A036681, A036682, A036683, A036684, A036692 (central terms).
Cf. A007318, A051159, A091533, A228196, A228576.
Some other Fibonacci-Pascal triangles: A027926, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

a036355 n k = a036355_tabl !! n !! k
a036355_row n = a036355_tabl !! n
a036355_tabl = [1] : f [1] [1,1] where
   f us vs = vs : f vs (zipWith (+)
                       (zipWith (+) ([0,0] ++ us) (us ++ [0,0]))
                       (zipWith (+) ([0] ++ vs) (vs ++ [0])))
-- Reinhard Zumkeller, Apr 23 2013

nmax = 11; t[n_, m_] := t[n, m] = tp[n-1, m-1] + tp[n-2, m-2] + tp[n-1, m] + tp[n-2, m]; tp[n_, m_] /; 0 <= m <= n && n >= 0 := t[n, m]; tp[n_, m_] = 0; t[0, 0] = 1; Flatten[ Table[t[n, m], {n, 0, nmax}, {m, 0, n}]] (* Jean-François Alcover, Nov 09 2011, after formula *)

/* same as in A092566 but use */
steps=[[1,0], [2,0], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */

T(n, m) = T'(n-1, m-1)+T'(n-2, m-2)+T'(n-1, m)+T'(n-2, m), where T'(n, m) = T(n, m) if 0<=m<=n and n >= 0 and T'(n, m)=0 otherwise. Initial term T(0, 0)=1.

G.f.: 1/(1-(1+y)*x-(1+y^2)*x^2). - Vladeta Jovovic, Oct 11 2003

A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 3, 5, 4, 1, 5, 12, 14, 8, 2, 8, 25, 38, 32, 15, 3, 13, 50, 94, 104, 71, 28, 5, 21, 96, 215, 293, 260, 149, 51, 8, 34, 180, 468, 756, 822, 612, 304, 92, 13, 55, 331, 980, 1828, 2346, 2136, 1376, 604, 164, 21

Offset: 0

Philippe Deléham, Jan 29 2012

Antidiagonal sums are in A052980, row sums are in A046717.

Similar to A091533 and to A091562. Triangle satisfying the same recurrence as A091533 and A091562, but with the initial values T(0,0) = 1, T(0,1) = 1, T(1,1) = 0.

			Triangle begins :
1
1, 0
2, 2, 1
3, 5, 4, 1
5, 12, 14, 8, 2
8, 25, 38, 32, 15, 3
13, 50, 94, 104, 71, 28, 5

Cf. Column 0: A000045, Diagonals : A000045, A029907, A036681.

Cf. A090171, A090172, A090173, A090174, A091533, A091562 (same recurrence).

T(n,k) = {if(n<0, return(0)); if (n==0, if (k<0, return(0)); if (k==0, return(1))); if (n==1, if (k<0, return(0)); if (k==0, return(1)); if (k==1, return(0))); T(n-1,k)+T(n-1,k-1)+T(n-2,k)+T(n-2,k-1)+T(n-2,k-2);} \\ Michel Marcus, Oct 27 2021

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2) for n>=2, k>=0, with initial conditions specified by first two rows. T(0,0) = 1, T(1,0) = 1, T(1,1) = 0.

a(46), a(48) corrected by Georg Fischer, Oct 27 2021

A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

Original entry on oeis.org

2, 7, 21, 53, 124, 273, 577, 1181, 2358, 4614, 8880, 16854, 31612, 58691, 108003, 197203, 357596, 644463, 1155059, 2059897, 3656988, 6465660, 11388480, 19990140, 34976870, 61019071, 106160481, 184228193, 318948124, 550962717, 949781269, 1634103701, 2806342578

Offset: 3

N. J. A. Sloane, May 09 2012

Apparently the number of Dyck n-paths that have n-2 peaks after changing each valley to a peak by the transformation DU -> UD. E.g., the Dyck 3-paths UUUDDD and UUDUDD have 1 peak after changing DU to UD so a(3) = 2. - David Scambler, Sep 03 2012

Robert P. P. McKone, Table of n, a(n) for n = 3..5000
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. Column 2 of A091533. Partial sums of A036681.

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 7, 21, 53}, {3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
A212338[1, n2_] = 0; A212338[n1_, 1] = 0; A212338[2, n2_] = 0; A212338[n1_, 2] = 0; A212338[3, 3] = 1; A212338[n1_, n2_] := A212338[n1, n2] = A212338[n1 - 1, n2] + A212338[n1, n2 - 1] + A212338[n1 - 1, n2 - 1] + A212338[n1 - 2, n2] + A212338[n1, n2 - 2]; Table[A212338[5, y], {y, 3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
QQQ2[t,x]=2/(1 + (t*x - t)*(1 + t) +((1 + (t*x - t)*(1 + t))^2 - 4*t*x)^(1/2)); CoefficientList[Coefficient[Series[QQQ2[t, x], {t, 0, 22}], x],t] (* Robert Price, Jun 05 2012 *)

g.f. -x^3*(2+x) / (x^2+x-1)^3, i.e., a(n) = 2*A001628(n-3) + A001628(n-4). - R. J. Mathar, Jun 27 2012

a(n) = a(n-1) + a(n-2) + A067331(n-3). E.g., a(5) = 21 = 7 + 2 + 12. - David Scambler, Sep 03 2012

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A036355 Fibonacci-Pascal triangle read by rows.

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A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

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A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

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