A120895 -id:A120895 - cp's OEIS frontend

A120894 Cascadence of 1+x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,1,1] 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, 1, 1, 1, 2, 3, 2, 1, 2, 5, 7, 6, 5, 3, 2, 5, 12, 18, 18, 14, 10, 10, 7, 5, 12, 30, 48, 50, 42, 34, 27, 22, 24, 17, 12, 30, 78, 128, 140, 126, 103, 83, 73, 63, 53, 59, 42, 30, 78, 206, 346, 394, 369, 312, 259, 219, 189, 175, 154, 131, 150, 108, 78, 206, 552, 946, 1109

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+G+G^2), so that G/x = g.f. of Motzkin numbers (A001006). 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;
1, 1, 1;
2, 3, 2, 1, 2;
5, 7, 6, 5, 3, 2, 5;
12, 18, 18, 14, 10, 10, 7, 5, 12;
30, 48, 50, 42, 34, 27, 22, 24, 17, 12, 30;
78, 128, 140, 126, 103, 83, 73, 63, 53, 59, 42, 30, 78;
206, 346, 394, 369, 312, 259, 219, 189, 175, 154, 131, 150, 108, 78, 206;
552, 946, 1109, 1075, 940, 790, 667, 583, 518, 460, 435, 389, 336, 392, 284, 206, 552;
1498, 2607, 3130, 3124, 2805, 2397, 2040, 1768, 1561, 1413, 1284, 1160, 1117, 1012, 882, 1042, 758, 552, 1498; ...
Convolution of [1,1,1] with each row produces:
[1,1,1]*[1] = [1,1,1];
[1,1,1]*[1,1,1] = [1,2,3,2,1];
[1,1,1]*[2,3,2,1,2] = [2,5,7,6,5,3,2];
[1,1,1]*[5,7,6,5,3,2,5] = [5,12,18,18,14,10,10,7,5];
[1,1,1]*[12,18,18,14,10,10,7,5,12] = [12,30,48,50,42,34,27,22,24,17,12]; ...
These convoluted rows, when concatenated, yield the sequence:
1,1,1, 1,2,3,2,1, 2,5,7,6,5,3,2, 5,12,18,18,14,10,10,7,5, ...
which equals the concatenated rows of this original triangle:
1, 1,1,1, 2,3,2,1,2, 5,7,6,5,3,2,5, 12,18,18,14,10,10,7,5,12, ...

Paul D. Hanna, Table of n, a(n) for n = 0..440

Cf. A120895 (column 0), A120896 (central terms), A120897 (row sums), A001006 (Motzkin numbers); variants: A092683, A092686, A120898.

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

/* Generated by the G.F.: */
{T(n,k)=local(A,F=1+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+x+x^2. - Paul D. Hanna, Jul 17 2006

A120899 G.f. satisfies: A(x) = C(x)^2 * A(x^3*C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 5, 16, 54, 186, 654, 2338, 8463, 30938, 114022, 423096, 1579049, 5922512, 22309350, 84354388, 320020227, 1217689680, 4645693038, 17766596202, 68092473570, 261486788434, 1005962436536, 3876412305114, 14960183283203

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Column 0 of triangle A120898 (cascadence of 1+2x+x^2). Self-convolution of A120900.

			A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 186*x^5 + 654*x^6 +...
= C(x)^2 * 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, A120900 (square-root), A120901, A120902; A000108; variants: A092684, A092687, A120895.

{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^2*subst(A,x,x^3*C^4 +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)}

A120897 Row sums of triangle A120894 (cascadence of 1+x+x^2).

Original entry on oeis.org

1, 3, 10, 33, 106, 336, 1056, 3296, 10234, 31648, 97551, 299888, 919865, 2816291, 8608712, 26278538, 80120533, 244022331, 742525242, 2257527861, 6858558246, 20822959508, 63181453350, 191601205342, 580749971840, 1759465640586

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Cf. A120894, A120895, A120896.

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)}

A120896 Central terms of triangle A120894 (cascadence of 1+x+x^2).

Original entry on oeis.org

1, 1, 2, 5, 10, 27, 73, 189, 518, 1413, 3857, 10707, 29804, 83365, 234665, 662654, 1878059, 5341302, 15231940, 43552610, 124829236, 358537455, 1031835260, 2974872009, 8590929683, 24847042484, 71964879243, 208704926659

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Cf. A120894, A120895, A120897.

cp's OEIS Frontend

A120894 Cascadence of 1+x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,1,1] 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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A120899 G.f. satisfies: A(x) = C(x)^2 * A(x^3*C(x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).

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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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A120897 Row sums of triangle A120894 (cascadence of 1+x+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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A120896 Central terms of triangle A120894 (cascadence of 1+x+x^2).

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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.