A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).

1, 3, 1, 5, 10, 1, 7, 35, 21, 1, 9, 84, 126, 36, 1, 11, 165, 462, 330, 55, 1, 13, 286, 1287, 1716, 715, 78, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1

Offset: 0

Author

Ralf Stephan, Feb 06 2005

Keywords

• nonn
• easy
• tabl

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

Examples

			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)
		

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.

Crossrefs

Reflected version of A091042. Cf. A086645, A103328.

Cf. A104033, A086645, A292219.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019

Magma

[Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

Mathematica

Flatten[Table[Binomial[2n+1,2k+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 19 2014 *)

Maxima

create_list(binomial(2*n+1,2*k+1),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

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

Sage

[[binomial(2*n+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

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

Sum_{k=0..n} T(n, k)*A000364(n-k) = A002084(n). - Philippe Deléham, Aug 27 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