A028364 -id:A028364 - cp's OEIS frontend

A028376 Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

0, 0, 1, 0, 1, 2, 0, 2, 3, 5, 0, 5, 7, 9, 14, 0, 14, 19, 23, 28, 42, 0, 42, 56, 66, 76, 90, 132, 0, 132, 174, 202, 227, 255, 297, 429, 0, 429, 561, 645, 715, 785, 869, 1001, 1430, 0, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 0, 4862, 6292, 7150

Offset: 0

Wouter Meeussen

			0;
0,1;
0,1,2;
0,2,3,5;
0,5,7,9,14;
0,14,19,23,28,42; ...

See A028364 for a better version. Cf. A028377, A028378.

A067325 Fourth column of triangle A067323.

Original entry on oeis.org

5, 19, 66, 227, 785, 2739, 9646, 34268, 122706, 442510, 1605956, 5861481, 21502585, 79243395, 293246550, 1089264360, 4059928950, 15179606010, 56917649820, 213982542150, 806429435994, 3046017513198

Offset: 0

Wolfdieter Lang, Feb 05 2002

Also fourth diagonal of triangle A028364.

Cf. A067324 (third column).

a(n)= A067323(n+3, 3)= C(n+4)-(C(n+3)+C(n+2)+2*C(n+1)), C(n) := A000108(n) (Catalan).

G.f.: (c(x)^3)*(2+2*c(x)+c(x)^2), with c(x) g.f. of A000108 (Catalan).

A067326 Fifth column of triangle A067323.

Original entry on oeis.org

14, 56, 202, 715, 2529, 8986, 32123, 115556, 418200, 1521976, 5567551, 20462525, 75528895, 279874350, 1040790135, 3883140600, 14531382060, 54529456320, 205146226200, 773608833894, 2923686178098, 11071970477876

Offset: 0

Wolfdieter Lang, Feb 05 2002

Also fifth diagonal of triangle A028364.

Cf. A067325 (fourth column).

a(n)= A067323(n+4, 4)= C(n+5)-sum(C(k)*C(n+4-k), k=0..3), C(n) := A000108(n) (Catalan).

G.f.: (c(x)^3)*(5+5*c(x)+3*c(x)^2+c(x)^3), with c(x) g.f. of A000108 (Catalan).

A073147 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A001764(n), a(n,n)=A006013(n), a(n,n-1)=A006629(n-1).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 12, 15, 18, 30, 55, 67, 76, 88, 143, 273, 328, 364, 400, 455, 728, 1428, 1701, 1866, 2010, 2175, 2448, 3876, 7752, 9180, 9999, 10659, 11319, 12138, 13566, 21318, 43263

Offset: 0

Paul D. Hanna, Jul 18 2002

Related to generalized Catalan numbers; in particular, C(3n,n)/(2n+1) (enumerates ternary trees and also non-crossing trees)(A001764) and sum of root degrees of all noncrossing trees on nodes on a circle (A006629).

These numbers are cardinalities of some intervals in the Tamari lattices. - F. Chapoton, Jul 15 2021

			{1}, {1,2}, {3,4,7}, {12,15,18,30}, {55,67,76,88,143}, {273,328,364,400,455,728},...

Cf. A001764, A006013, A006629, A028364.

(n, m)-th entry in triangle is Sum A001764(n-k)*A001764(k), k=0..m.

A073148 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A006013(n), a(n+1,n)=A001764(n+1), a(n,m) = Sum A001764(n-k)*a(n,k), k=0..m.

Original entry on oeis.org

1, 2, 3, 7, 9, 12, 30, 37, 43, 55, 143, 173, 194, 218, 273, 728, 871, 961, 1045, 1155, 1428, 3876, 4604, 5033, 5393, 5778, 6324, 7752, 21318, 25194, 27378, 29094, 30744, 32655, 35511, 43263, 120175

Offset: 0

Paul D. Hanna, Jul 18 2002

Compare to A073147. Related to generalized Catalan numbers; in particular, C(3n,n)/(2n+1) (enumerates ternary trees and also non-crossing trees)(A001764).

These numbers are cardinalities of some intervals in the Tamari lattices. - F. Chapoton, Jul 15 2021

			{1}, {2,3}, {7,9,12}, {30,37,43,55}, {143,173,194,218,273},{728,871,961,1045,1155,1428}, {3876,4604,5033,5393,5778,6324,7752}, ...

Cf. A073147, A001764, A006013, A006629, A028364.

a(n, m) = Sum A001764(n-k)*a(n, k), k=0..m.

A116871 Sixth column of triangle A067323.

Original entry on oeis.org

42, 174, 645, 2333, 8398, 30275, 109550, 398180, 1453908, 5332407, 19639521, 72616727, 269473750, 1003347975, 3747412770, 14036374680, 52714429260, 198459284400, 748867149234, 2831788492218, 10729442739596

Offset: 0

Wolfdieter Lang, Mar 24 2006

Also sixth diagonal sequence of triangle A028364.

Cf. A067326 (fifth column of A067323).

a(n)= A067323(n+5,5), n>=0.
a(n)= A028364(5+n,n) = sum(C(k)*C(5+n-k),k=0..n), n>=0, with the Catalan numbers C(n):=A000108(n).
G.f.: (c(x)^3)sum(C(4, k)*c(x)^k, k=0..4), with C(n, m) := (m+1)*binomial(2*n-m, n-m)/(n+1) (Catalan convolutions A033184).

cp's OEIS Frontend

A028376 Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

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A067325 Fourth column of triangle A067323.

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A067326 Fifth column of triangle A067323.

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A073147 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A001764(n), a(n,n)=A006013(n), a(n,n-1)=A006629(n-1).

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A073148 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A006013(n), a(n+1,n)=A001764(n+1), a(n,m) = Sum A001764(n-k)*a(n,k), k=0..m.

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A116871 Sixth column of triangle A067323.

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