A121399 -id:A121399 - cp's OEIS frontend

A121398 Main diagonal of triangle A121400; also equals the partial sums of column 0 (A121399) of the same triangle.

1, 2, 5, 11, 28, 70, 184, 486, 1313, 3576, 9851, 27319, 76286, 214120, 603858, 1709719, 4857959, 13845948, 39572583, 113380652, 325576692, 936796592, 2700456452, 7797587816, 22550434989, 65308288346, 189388557677

Offset: 0

Paul D. Hanna, Jul 27 2006

nonn

Cf. A121400 (triangle), A121399 (column 0), A001006 (Motzkin).

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

G.f. A(x) = A(x^2*G)*G*(1-x^2*G)/(1-x), where G(x) is the g.f. of the Motzkin numbers (A001006): G = (1 + x*G + x^2*G^2).

A121400 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for n>=k>=1, with T(0,0) = 1, T(n,n) = T(n,0) + T(n-1,n-1) for n>=1; T(n,k)=0 when n

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 11, 8, 11, 17, 25, 30, 19, 28, 42, 72, 74, 77, 47, 70, 114, 188, 223, 198, 194, 117, 184, 302, 525, 609, 615, 509, 495, 301, 486, 827, 1436, 1749, 1733, 1619, 1305, 1282, 787, 1313, 2263, 4012, 4918, 5101, 4657, 4206, 3374, 3382, 2100, 3576

Offset: 0

Paul D. Hanna, Jul 27 2006

Main diagonal (A121398) forms the partial sums of column 0 (A121399). The g.f. of the row sums is H(x)*(1-x)/(1-3x), where H(x) is the g.f. of column 0. This is the cascadence for function F(x) = 1 + x + x^2 when forced to form a triangle in which row n has n+1 terms for n>=0.

			Triangle begins:
1;
1, 2;
3, 3, 5;
6, 11, 8, 11;
17, 25, 30, 19, 28;
42, 72, 74, 77, 47, 70;
114, 188, 223, 198, 194, 117, 184;
302, 525, 609, 615, 509, 495, 301, 486;
827, 1436, 1749, 1733, 1619, 1305, 1282, 787, 1313;
2263, 4012, 4918, 5101, 4657, 4206, 3374, 3382, 2100, 3576;
6275, 11193, 14031, 14676, 13964, 12237, 10962, 8856, 9058, 5676, 9851;
The convolution of each row with [1,1,1] yields:
[1,1,1]*[1] = [1,1,1];
[1,1,1]*[1,2] = [1,3,3,2];
[1,1,1]*[3,3,5] = [3,6,11,8,5];
[1,1,1]*[6,11,8,11] = [6,17,25,30,19,11]; ...
Concatenate these convoluted rows after adding last and first terms:
1,1,1 + 1,3,3,2 + 3,6,11,8,5 + 6,17,25,30,19,11 + 17, ...
to obtain the concatenated rows of this original triangle:
1, 1,2, 3,3,5, 6,11,8,11, 17,25,30,19,28, ...

Cf. A121398 (main diagonal), A121399 (column 0), A001006 (Motzkin).

PARI
```
{T(n,k)=if(n
				
```

G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+y+y^2) - y ), where H(x) satisfies: H(x) = G*H(x*G)/x = g.f. of column 0 (A121399) and G/x is the g.f. of the Motzkin numbers (A001006): G = x*(1 + G + G^2).

cp's OEIS Frontend

A121398 Main diagonal of triangle A121400; also equals the partial sums of column 0 (A121399) of the same triangle.

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