A120899 -id:A120899 - cp's OEIS frontend

A120898 Cascadence of 1+2x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,2,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 2n+1 terms remain in row n for n>=0.

1, 2, 1, 2, 5, 6, 5, 2, 5, 16, 22, 18, 14, 12, 5, 16, 54, 78, 72, 58, 43, 38, 37, 16, 54, 186, 282, 280, 231, 182, 156, 128, 123, 124, 54, 186, 654, 1030, 1073, 924, 751, 622, 535, 498, 425, 418, 426, 186, 654, 2338, 3787, 4100, 3672, 3048, 2530, 2190, 1956, 1766

Offset: 0

Paul D. Hanna, Jul 14 2006

In this case, the g.f. of column 0, H(x), satisfies: H(x) = H(x*G^2)*G/x where G satisfies: G = x*(1+2G+G^2), so that 1+G = g.f. of Catalan numbers (A000108). More generally, the cascadence of polynomial F(x) of degree d, F(0)=1, is a triangle with d*n+1 terms in row n where the g.f. H(x) of column 0 satisfies: H(x) = H(x*G^d)*G/x where G = x*F(G); thus G = series_reversion(x/F(x)), or, equivalently, [x^n] G = [x^n] x*F(x)^n/n for n>=1.

Further, the g.f. of the cascadence triangle for polynomial F(x) of degree d is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) = G*H(x*G^d)/x and G = x*F(G). - Paul D. Hanna, Jul 17 2006

			Triangle begins:
1;
2, 1, 2;
5, 6, 5, 2, 5;
16, 22, 18, 14, 12, 5, 16;
54, 78, 72, 58, 43, 38, 37, 16, 54;
186, 282, 280, 231, 182, 156, 128, 123, 124, 54, 186;
654, 1030, 1073, 924, 751, 622, 535, 498, 425, 418, 426, 186, 654;
2338, 3787, 4100, 3672, 3048, 2530, 2190, 1956, 1766, 1687, 1456, 1452, 1494, 654, 2338; ...
Convolution of [1,2,1] with each row produces:
[1,2,1]*[1] = [1,2,1];
[1,2,1]*[2,1,2] = [2,5,6,5,2];
[1,2,1]*[5,6,5,2,5] = [5,16,22,18,14,12,5];
[1,2,1]*[16,22,18,14,12,5,16] = [16,54,78,72,58,43,38,37,16];
These convoluted rows, when concatenated, yield the sequence:
1,2,1, 2,5,6,5,2, 5,16,22,18,14,12,5, 16,54,78,72,58,43,38,37,16, ...
which equals the concatenated rows of this original triangle:
1, 2,1,2, 5,6,5,2,5, 16,22,18,14,12,5,16, 54,78,72,58,43,38,37,16,54,

Paul D. Hanna, Table of n, a(n) for n = 0..440; rows 0..20 of flattened triangle.

Cf. A120899 (column 0), A120901 (central terms), A120902 (row sums), A000108 (Catalan); variants: A092683, A092686, A120894.

PARI
```
T(n,k)=if(2*n
				
```

/* Generated by the G.F.: */
{T(n,k)=local(A,F=1+2*x+x^2,d=2,G=x,H=1+x,S=ceil(log(n+1)/log(d+1))); for(i=0,n,G=x*subst(F,x,G+x*O(x^n)));for(i=0,S,H=subst(H,x,x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H,x,x*y^d +x*O(x^n)))/(x*subst(F,x,y)-y); polcoeff(polcoeff(A,n,x),k,y)}
for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jul 17 2006

G.f.: A(x,y) = ( x*H(x) - y*H(x*y^2) )/( x*F(y) - y ), where H(x) = G*H(x*G^2)/x, G = x*F(G), F(x)=1+2x+x^2. - Paul D. Hanna, Jul 17 2006

A120895 G.f. satisfies: A(x) = G(x)A(x^3G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

Original entry on oeis.org

1, 1, 2, 5, 12, 30, 78, 206, 552, 1498, 4105, 11340, 31541, 88237, 248076, 700478, 1985397, 5646129, 16104378, 46056513, 132031176, 379315946, 1091890772, 3148736064, 9095091878, 26310816944, 76219704957, 221085782559, 642058752476, 1866693825362, 5432795508417

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Equals column 0 and main diagonal of triangle A120894 (cascadence of 1+x+x^2).

			A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 78*x^6 + 206*x^7+...
= G(x)*A(x^3*G(x)^2) where
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...
is the g.f. of the Motzkin numbers (A001006) so that G(x) satisfies:
G(x) = 1 + x*G(x) + x^2*G(x)^2.

Cf. A120894, A120896, A120897; A001006; variants: A092684, A092687, A120899.

{a(n)=local(A=1+x,G=1/x*serreverse(x/(1+x+x^2+x*O(x^n)))); for(i=0,n,A=G*subst(A,x,x^3*G^2 +x*O(x^n)));polcoeff(A,n,x)}

A120920 G.f. satisfies: A(x) = G(x)^3 * A(x^4G(x)^9), where G(x) is the g.f. of the number of ternary trees (A001764): G(x) = 1 + xG(x)^3.

Original entry on oeis.org

1, 3, 12, 55, 276, 1464, 8058, 45543, 262626, 1538607, 9130446, 54761628, 331403447, 2021021082, 12407102937, 76611488305, 475493441604, 2964664310319, 18560063203353, 116621922800283, 735236268006654

Offset: 0

Paul D. Hanna, Jul 17 2006

nonn

Column 0 of triangle A120919 (cascadence of (1+x)^3).

			A(x) = 1 + 3*x + 12*x^2 + 55*x^3 + 276*x^4 + 1464*x^5 + 8058*x^6 +...
= G(x)^3 * A(x^4*G(x)^9) where
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
is g.f. of A001764: G(x) = 1 + x*G(x)^3.

Cf. A120919, A120921 (cube-root), A120922, A120923; A001764; variants: A092684, A092687, A120895, A120899, A120920.

{a(n)=local(A=1+x,G=(1/x*serreverse(x/(1+3*x+3*x^2+x^3+x*O(x^n))))^(1/3)); for(i=0,n,A=G^3*subst(A,x,x^4*G^9 +x*O(x^n)));polcoeff(A,n,x)}

A120902 Row sums of triangle A120898 (cascadence of 1+2x+x^2).

Original entry on oeis.org

1, 5, 23, 103, 450, 1932, 8196, 34468, 143997, 598463, 2476936, 10216818, 42023225, 172436363, 706132290, 2886574198, 11781962631, 48025519977, 195530083266, 795241236228, 3231290822280, 13118557603984, 53218706064038

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Cf. A120898, A120899, A120900, A120901.

A120915 G.f. satisfies: A(x) = C(2x)^2 * A(x^3*C(2x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 20, 116, 720, 4656, 30996, 210896, 1459536, 10239796, 72651184, 520328112, 3756512912, 27307671040, 199705789248, 1468209751856, 10844681408064, 80437588353600, 598867568439828, 4473784063109904, 33524058847464912

Offset: 0

Paul D. Hanna, Jul 17 2006

nonn

Column 0 of triangle A120914 (cascadence of (1+2x)^2).

			A(x) = 1 + 4*x + 20*x^2 + 116*x^3 + 720*x^4 + 4656*x^5 + 30996*x^6 +...
= C(2x)^2 * A(x^3*C(2x)^4) where
C(2x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + 8448*x^6 +...
and C(x) is g.f. of the Catalan numbers (A000108): C(x) = 1 + x*C(x)^2.

Cf. A120914, A120916 (square-root), A120917, A120918; A000108; variants: A092684, A092687, A120895, A120899, A120920.

{a(n)=local(A=1+x,C=(1/x*serreverse(x/(1+4*x+4*x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C^2*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}

A120900 G.f. satisfies: A(x) = C(x)A(x^3C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 2, 6, 19, 62, 209, 722, 2539, 9054, 32654, 118876, 436171, 1611067, 5984943, 22344455, 83786875, 315397144, 1191324649, 4513742858, 17149228138, 65318912291, 249356597492, 953902701488, 3656057618727, 14037222220896

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Self-convolution equals A120899, which equals column 0 of triangle A120898 (cascadence of 1+2x+x^2).

			A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 62*x^5 + 209*x^6 + 722*x^7 +...
= C(x) * A(x^3*C(x)^4) where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
is the g.f. of the Catalan numbers (A000108): C(x) = 1 + x*C(x)^2.

Cf. A120898, A120899, A000108.

{a(n)=local(A=1+x,C=(1/x*serreverse(x/(1+2*x+x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}

A120901 Central terms of triangle A120898 (cascadence of 1+2x+x^2).

Original entry on oeis.org

1, 1, 5, 14, 43, 156, 535, 1956, 7175, 26418, 98375, 367176, 1378022, 5193625, 19641164, 74535167, 283651169, 1082274210, 4139129734, 15863315213, 60913982404, 234317240601, 902804442380, 3483620505111, 13460665855850

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Cf. A120898, A120899, A120900, A120902.

cp's OEIS Frontend

A120898 Cascadence of 1+2x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,2,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 2n+1 terms remain in row n for n>=0.

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A120895 G.f. satisfies: A(x) = G(x)*A(x^3*G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

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A120920 G.f. satisfies: A(x) = G(x)^3 * A(x^4*G(x)^9), where G(x) is the g.f. of the number of ternary trees (A001764): G(x) = 1 + x*G(x)^3.

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A120902 Row sums of triangle A120898 (cascadence of 1+2x+x^2).

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A120915 G.f. satisfies: A(x) = C(2x)^2 * A(x^3*C(2x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

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A120900 G.f. satisfies: A(x) = C(x)*A(x^3*C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

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A120901 Central terms of triangle A120898 (cascadence of 1+2x+x^2).

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A120895 G.f. satisfies: A(x) = G(x)A(x^3G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

A120920 G.f. satisfies: A(x) = G(x)^3 * A(x^4G(x)^9), where G(x) is the g.f. of the number of ternary trees (A001764): G(x) = 1 + xG(x)^3.

A120900 G.f. satisfies: A(x) = C(x)A(x^3C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).