cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A092686 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, when flattened, equals this flattened form of the original triangle.

Original entry on oeis.org

1, 2, 2, 6, 4, 6, 16, 14, 12, 16, 46, 40, 40, 32, 46, 132, 120, 112, 110, 92, 132, 384, 352, 334, 312, 316, 264, 384, 1120, 1038, 980, 940, 896, 912, 768, 1120, 3278, 3056, 2900, 2776, 2704, 2592, 2656, 2240, 3278, 9612, 9012, 8576, 8256, 8000, 7840, 7552, 7758
Offset: 0

Views

Author

Paul D. Hanna, Mar 04 2004

Keywords

Comments

First column and main diagonal forms A092687. Row sums form A092688.
This triangle is the cascadence of binomial (1+2x). 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. of the triangle, A(x,y), is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) satisfies: H(x) = G*H(x*G^d)/x and G=G(x) satisfies: G(x) = x*F(G(x)) so that G = series_reversion(x/F(x)); also, H(x) is the g.f. of column 0. - Paul D. Hanna, Jul 17 2006

Examples

			Rows begin:
1;
2, 2;
6, 4, 6;
16, 14, 12, 16;
46, 40, 40, 32, 46;
132, 120, 112, 110, 92, 132;
384, 352, 334, 312, 316, 264, 384;
1120, 1038, 980, 940, 896, 912, 768, 1120;
3278, 3056, 2900, 2776, 2704, 2592, 2656, 2240, 3278;
9612, 9012, 8576, 8256, 8000, 7840, 7552, 7758, 6556, 9612;
28236, 26600, 25408, 24512, 23840, 23232, 22862, 22072, 22724, 19224, 28236; ...
Convolution of each row with {1,2} results in the triangle:
1, 2;
2, 6, 4;
6, 16, 14, 12;
16, 46, 40, 40, 32;
46, 132, 120, 112, 110, 92;
132, 384, 352, 334, 312, 316, 264;
384, 1120, 1038, 980, 940, 896, 912, 768; ...
which, when flattened, equals the original triangle in flattened form.

Links

Crossrefs

Cf. A092683, A092687, A092688, A092689, A120894, A120898.

Programs

  • PARI
    T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,2, if(k==n,T(n,0), 2*T(n-1,k)+T(n-1,k+1)))))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
  • PARI
    /* Generate Triangle by the G.F.: */
{T(n,k)=local(A,F=1+2*x,d=1,G=x,H=1+2*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, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jul 17 2006

Formula

T(n, k) = 2*T(n-1, k) + T(n-1, k+1) for 0<=k
G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+2y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-2x))/(1-2x) and H(x) is the g.f. of column 0 (A092687). - 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

Views

Author

Paul D. Hanna, Jul 14 2006

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A120898, A120900 (square-root), A120901, A120902; A000108; variants: A092684, A092687, A120895.

Programs

  • PARI
    {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)}

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).

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

Views

Author

Paul D. Hanna, Jul 14 2006

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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^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.

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

Views

Author

Paul D. Hanna, Jul 17 2006

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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)}

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

Views

Author

Paul D. Hanna, Jul 17 2006

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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)}

A092688 Row sums of triangle A092686, in which the convolution of each row with {1,2} produces a triangle that, when flattened, equals the flattened form of A092686.

Original entry on oeis.org

1, 4, 16, 58, 204, 698, 2346, 7774, 25480, 82774, 266946, 855674, 2728702, 8663402, 27400862, 86376186, 271488444, 851099874, 2661967502, 8308462182, 25883429326, 80497346294, 249956869434, 775048966478, 2400067860090
Offset: 0

Views

Author

Paul D. Hanna, Mar 04 2004

Keywords

Crossrefs

Cf. A092683, A092686, A092687, A092689.

Programs

  • PARI
    {T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,2, if(k==n,T(n,0), 2*T(n-1,k)+T(n-1,k+1)))))}
a(n)=sum(k=0,n,T(n,k))
  • PARI
    {a(n)=local(A,F=1+2*x,d=1,G=x,H=1+2*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); sum(k=0,d*n,polcoeff(polcoeff(A,n,x),k,y))} \\ Paul D. Hanna, Jul 17 2006

Formula

G.f.: A(x) = H(x)*(1-x)/(1-3*x), where H(x) satisfies: H(x) = H(x^2/(1-2x))/(1-2x) and H(x) is the g.f. of A092687. - Paul D. Hanna, Jul 17 2006
Showing 1-6 of 6 results.

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