A268647 G.f.: C(x,y)^2 - S(x,y)^2 = Sum_{n>=0} x^(2*n)*y/[Sum_{k=0..2*n+1} T(n,k)*y^k], where C(x,y) = Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (k + y) and S(x,y) = Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (k + y).

0, 1, 2, 5, 4, 1, 48, 124, 120, 55, 12, 1, 2160, 6012, 6636, 3829, 1260, 238, 24, 1, 161280, 478656, 582080, 387260, 157080, 40593, 6720, 690, 40, 1, 18144000, 56772000, 74396520, 54801076, 25494150, 7927205, 1690920, 248523, 24750, 1595, 60, 1, 2874009600, 9397658880, 13075800192, 10415648880, 5357255904, 1893627736, 476011536, 86550035, 11423412, 1084083, 72072, 3185, 84, 1, 610248038400, 2071437822720, 3028563232128, 2569081620624, 1429040500160, 556365173000, 157528627256, 33179499353, 5260335080, 629597540, 56560504, 3753022, 178360, 5740, 112, 1

Offset

0,3

Comments

This triangle illustrates the following identity.

Given

C(x,y) = Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (k + y)

S(x,y) = Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (k + y)

then

C(x,y)^2 - S(x,y)^2 = Sum_{n>=0} x^(2*n) * y / ((n + y) * Product_{k=1..2*n} (k + y)).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..991 terms in rows 0..30 of this triangle in flattened form.

Formula

G.f. of row n: (n + y) * Product_{k=1..2*n} (k + y) = Sum_{k=0..2*n+1} T(n,k)*y^k, for n>=0.

Row sums equal A002674 (with offset): A002674(n+1) = (n+1)*(2*n+1)!.

Example

Define C(x,y) by the series:
C(x,y) = 1 + x^2/((1+y)*(2+y)) + x^4/((1+y)*(2+y)*(3+y)*(4+y)) + x^6/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)) + x^8/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)) +...
and define S(x,y) by the series:
S(x,y) = x/(1+y) + x^3/((1+y)*(2+y)*(3+y)) + x^5/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)) + x^7/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)) + x^9/((1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)*(9+y)) +...
then the g.f. of this triangle begins:
C(x,y)^2 - S(x,y)^2 = 1 + x^2*y/((1+y) * (1+y)*(2+y)) + x^4*y/((2+y) * (1+y)*(2+y)*(3+y)*(4+y)) + x^6*y/((3+y) * (1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)) + x^8*y/((4+y) * (1+y)*(2+y)*(3+y)*(4+y)*(5+y)*(6+y)*(7+y)*(8+y)) +...
where the rows of this triangle are formed from the coefficients in the denominators of coefficients of x^(2*n) in C(x,y)^2 - S(x,y)^2, as more clearly seen in the expansion:
C(x,y)^2 - S(x,y)^2 = y/(0 + y) + x^2 * y/(2 + 5*y + 4*y^2 + y^3) +
x^4 * y/(48 + 124*y + 120*y^2 + 55*y^3 + 12*y^4 + y^5) +
x^6 * y/(2160 + 6012*y + 6636*y^2 + 3829*y^3 + 1260*y^4 + 238*y^5 + 24*y^6 + y^7) +
x^8 * y/(161280 + 478656*y + 582080*y^2 + 387260*y^3 + 157080*y^4 + 40593*y^5 + 6720*y^6 + 690*y^7 + 40*y^8 + y^9) +...
This triangle begins:
0, 1;
2, 5, 4, 1;
48, 124, 120, 55, 12, 1;
2160, 6012, 6636, 3829, 1260, 238, 24, 1;
161280, 478656, 582080, 387260, 157080, 40593, 6720, 690, 40, 1;
18144000, 56772000, 74396520, 54801076, 25494150, 7927205, 1690920, 248523, 24750, 1595, 60, 1;
2874009600, 9397658880, 13075800192, 10415648880, 5357255904, 1893627736, 476011536, 86550035, 11423412, 1084083, 72072, 3185, 84, 1;
610248038400, 2071437822720, 3028563232128, 2569081620624, 1429040500160, 556365173000, 157528627256, 33179499353, 5260335080, 629597540, 56560504, 3753022, 178360, 5740, 112, 1; ...

Prog

(PARI) /* C(x, y)^2 - S(x, y)^2 = Sum_{n>=0} x^(2*n)*y/[Sum_{k=0..2*n+1} T(n, k)*y^k] */
{T(n, k) = my(C=1, S=x); C = sum(m=0, n+1, x^(2*m)/prod(k=1, 2*m, k + y) +x*O(x^(2*n)));
S = sum(m=1, n+1, x^(2*m-1)/prod(k=1, 2*m-1, k + y) +x*O(x^(2*n)));
polcoeff( y/polcoeff( C^2 - S^2, 2*n, x), k, y)}
for(n=0, 10, for(k=0, 2*n+1, print1(T(n, k), ", ")); print(""))
(PARI) /* (n + y)*Product_{k=1..2*n} (k + y) = Sum_{k=0..2*n+1} T(n, k)*y^k */
{T(n, k) = polcoeff((n + y)*prod(k=1, 2*n, k + y), k, y)}
for(n=0, 10, for(k=0, 2*n+1, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A322627 (diagonal).

Sequence in context: A087561 A309771 A009738 * A177067 A055127 A152669

Adjacent sequences: A268644 A268645 A268646 * A268648 A268649 A268650

Keyword

nonn,tabf

Author

Paul D. Hanna, Mar 01 2016

Status

approved