A092689 -id:A092689 - cp's OEIS frontend

A092683 Triangle, read by rows, such that the convolution of each row with {1,1} produces a triangle which, when flattened, equals this flattened form of the original triangle.

1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 6, 5, 5, 3, 6, 11, 10, 8, 9, 6, 11, 21, 18, 17, 15, 17, 11, 21, 39, 35, 32, 32, 28, 32, 21, 39, 74, 67, 64, 60, 60, 53, 60, 39, 74, 141, 131, 124, 120, 113, 113, 99, 113, 74, 141, 272, 255, 244, 233, 226, 212, 212, 187, 215, 141, 272, 527, 499

Offset: 0

Paul D. Hanna, Mar 04 2004

First column and main diagonal forms A092684. Row sums form A092685.

This triangle is the cascadence of binomial (1+x). 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

			Rows begin:
1;
1, 1;
2, 1, 2;
3, 3, 2, 3;
6, 5, 5, 3, 6;
11, 10, 8, 9, 6, 11;
21, 18, 17, 15, 17, 11, 21;
39, 35, 32, 32, 28, 32, 21, 39;
74, 67, 64, 60, 60, 53, 60, 39, 74;
141, 131, 124, 120, 113, 113, 99, 113, 74, 141;
272, 255, 244, 233, 226, 212, 212, 187, 215, 141, 272;
527, 499, 477, 459, 438, 424, 399, 402, 356, 413, 272, 527;
1026, 976, 936, 897, 862, 823, 801, 758, 769, 685, 799, 527, 1026; ...
The convolution of each row with {1,1} gives the triangle:
1, 1;
1, 2, 1;
2, 3, 3, 2;
3, 6, 5, 5, 3;
6, 11, 10, 8, 9, 6;
11, 21, 18, 17, 15, 17, 11;
21, 39, 35, 32, 32, 28, 32, 21;
39, 74, 67, 64, 60, 60, 53, 60, 39; ...
which, when flattened, equals the original triangle in flattened form.

Paul D. Hanna, Table of n, a(n) for n = 0..1325; rows 0..50 in flattened form.

Cf. A092684, A092685, A092686, A092689, A120894, A120898.

T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), T(n-1,k)+T(n-1,k+1)))))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

/* Generate Triangle by G.F. where F=1+x: */
{T(n,k)=local(A,F=1+x,d=1,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, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jul 17 2006

T(n, k) = 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+y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-x))/(1-x) and H(x) is the g.f. of column 0 (A092684). - Paul D. Hanna, Jul 17 2006

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

Paul D. Hanna, Mar 04 2004

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

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..495

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

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(""))

/* 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

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

A092684 First column and main diagonal of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 39, 74, 141, 272, 527, 1026, 2002, 3914, 7659, 14996, 29369, 57531, 112727, 220963, 433342, 850386, 1670011, 3282259, 6456475, 12711413, 25047465, 49396116, 97490480, 192552549, 380565123, 752619506, 1489234257

Offset: 0

Paul D. Hanna, Mar 04 2004

nonn

The self-convolution forms A100938. - Paul D. Hanna, Nov 23 2004

The limit of the matrix power A011973^n, as n->inf, results in a single column vector equal to this sequence. - Paul D. Hanna, May 03 2006

			a(8) = Sum_{k=0..[8/2]} C(n-k,k)*a(k)
= C(8,0)*a(0) +C(7,1)*a(1) +C(6,2)*a(2) +C(5,3)*a(3) +C(4,4)*a(4)
= 1*1 + 7*1 + 15*2 + 10*3 + 1*6 = 74.

Cf. A092683, A092685, A092686, A092689.

Cf. A011973 (Fibonacci polynomials), A100938 (self-convolution).

{T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), T(n-1,k)+T(n-1,k+1)))))}
a(n)=T(n,0)

a(n)=if(n==0,1,sum(k=0,n\2,binomial(n-k,k)*a(k))) \\ Paul D. Hanna, May 03 2006

{a(n)=local(A=1+x);for(i=0,n\2,A=subst(A,x,x^2/(1-x+x*O(x^n)))/(1-x));polcoeff(A,n)} \\ Paul D. Hanna, Jul 10 2006

Invariant under the transformation of Fibonacci triangle A011973(n,k)=C(n-k,k): a(n) = Sum_{k=0..[n/2]} C(n-k,k)*a(k). - Paul D. Hanna, May 03 2006
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*a(k). - Vladeta Jovovic, May 07 2006
G.f. satisfies: A(x) = A( x^2/(1-x) )/(1-x). - Paul D. Hanna, Jul 10 2006

A092687 First column and main diagonal 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, 2, 6, 16, 46, 132, 384, 1120, 3278, 9612, 28236, 83072, 244752, 722048, 2132704, 6306304, 18666190, 55300732, 163968612, 486528288, 1444571068, 4291629384, 12756459936, 37934818112, 112855778768, 335867740704, 999895548736

Offset: 0

Paul D. Hanna, Mar 04 2004

nonn

Conjecture: Limit n->infinity a(n)^(1/n) = 3. - Vaclav Kotesovec, Jun 29 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..850

Cf. A092683, A092686, A092688, A092689.

m = 27; A[] = 1; Do[A[x] = A[x^2/(1-2x)]/(1-2x) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)

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)=T(n,0)
for(n=0,30,print1(a(n),", "))

a(n)=local(A=1+x);for(i=0,n\2,A=subst(A,x,x^2/(1-2*x+x*O(x^n)))/(1-2*x));polcoeff(A,n) \\ Paul D. Hanna, Jul 10 2006

/* Using Recurrence: */
a(n)=if(n==0, 1, sum(k=0, n\2, binomial(n-k, k)*2^(n-2*k)*a(k)))
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jul 10 2006

G.f. satisfies: A(x) = A( x^2/(1-2x) )/(1-2x). Recurrence: a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*2^(n-2k)*a(k). - Paul D. Hanna, Jul 10 2006

A092685 Row sums of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.

Original entry on oeis.org

1, 2, 5, 11, 25, 55, 120, 258, 551, 1169, 2469, 5193, 10885, 22746, 47404, 98553, 204443, 423259, 874680, 1804556, 3717348, 7647075, 15711194, 32242013, 66096274, 135366764, 276988466, 566312984, 1156974619, 2362043602

Offset: 0

Paul D. Hanna, Mar 04 2004

nonn

Cf. A092683, A092684, A092686, A092689, A120897, A120902.

{T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), T(n-1,k)+T(n-1,k+1)))))}
a(n)=sum(k=0,n,T(n,k))

{a(n)=local(A,F=1+x,d=1,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); sum(k=0,2*n,polcoeff(polcoeff(A,n,x),k,y))} \\ Paul D. Hanna, Jul 17 2006

G.f.: A(x,y) = H(x)*(1-x)/(1-2*x), where H(x) satisfies: H(x) = H(x^2/(1-x))/(1-x) and H(x) is the g.f. of A092684. - Paul D. Hanna, Jul 17 2006

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

Paul D. Hanna, Mar 04 2004

nonn

Cf. A092683, A092686, A092687, A092689.

{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))

{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

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

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A092690 Row sums of triangle A092689, which is related to the central trinomial coefficients (A002426).

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A092683 Triangle, read by rows, such that the convolution of each row with {1,1} produces a triangle which, when flattened, equals this flattened form of the original triangle.

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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.

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A092684 First column and main diagonal of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.

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A092687 First column and main diagonal 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.

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A092685 Row sums of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.

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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.

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