OFFSET
0,2
COMMENTS
A subset of Pascal's triangle A007318.
Elements have the same parity as those of Pascal's triangle.
Matrix inverse is A104033. - Paul D. Hanna, Feb 28 2005
Row reverse of A091042. - Peter Bala, Jul 29 2013
Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. Cf. A086645. - Peter Bala, Aug 06 2013
The row polynomial P(d, x) = Sum_{k=0..d} T(d, k)*x^k, multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,3] given in A292219. - Wolfdieter Lang, Oct 12 2017
REFERENCES
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
LINKS
Indranil Ghosh, Rows 0..120 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
FORMULA
G.f. for column k: Sum_{j=0..k+1} C(2*(k+1), 2*j)x^j)/(1-x)^(2*(k+1)). - Paul Barry, Feb 24 2005
G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - Paul D. Hanna, Feb 28 2005
E.g.f.: 1/sqrt(x)*sinh(sqrt(x)*t)*cosh(t) = t + (3 + x)*t^3/3! + (5 + 10*x + x^2)*t^5/5! + .... - Peter Bala, Jul 29 2013
T(n+2,k+2) = 2*T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k+2) + 2*T(n,k+1) - T(n,k). - Emanuele Munarini, Jul 05 2017
EXAMPLE
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 3 1
2: 5 10 1
3: 7 35 21 1
4: 9 84 126 36 1
5: 11 165 462 330 55 1
6: 13 286 1287 1716 715 78 1
7: 15 455 3003 6435 5005 1365 105 1
8: 17 680 6188 19448 24310 12376 2380 136 1
9: 19 969 11628 50388 92378 75582 27132 3876 171 1
10: 21 1330 20349 116280 293930 352716 203490 54264 5985 210 1
... reformatted and extended. - Wolfdieter Lang, Oct 12 2017
From Peter Bala, Aug 06 2013: (Start)
Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n+1)! the column generating functions begin
1st col: cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + 9*y^4/9! + ....
2nd col: 1/3!*y*cosh(sqrt(y)) = y/3! + 10*y^2/5! + 35*y^3/7! + 84*y^4/9! + ....
3rd col: 1/5!*y^2*cosh(sqrt(y)) = y^2/5! + 21*y^3/7!! + 126*y^4/9! + 462*y^5/11! + .... (End)
MATHEMATICA
Flatten[Table[Binomial[2n+1, 2k+1], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jun 19 2014 *)
PROG
(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1+X*(1-Y))/((1+X*(1-Y))^2-4*X), n, x), k, y)} \\ Paul D. Hanna, Feb 28 2005
(PARI) T(n, k) = binomial(2*n+1, 2*k+1);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019
(Maxima) create_list(binomial(2*n+1, 2*k+1), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */
(Magma) [Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019
(Sage) [[binomial(2*n+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019
CROSSREFS
KEYWORD
AUTHOR
Ralf Stephan, Feb 06 2005
STATUS
approved