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A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1). 11
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 (list; table; graph; refs; listen; history; text; internal format)
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 is sum{j = 0..k+1, C(2(k+1), 2j)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<=k<=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]

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

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

CROSSREFS

Reflected version of A091042. Cf. A086645, A103328.

Cf. A104033. A086645. A292219.

Sequence in context: A146255 A122366 A228781 * A177463 A065229 A233037

Adjacent sequences:  A103324 A103325 A103326 * A103328 A103329 A103330

KEYWORD

nonn,easy,tabl

AUTHOR

Ralf Stephan, Feb 06 2005

STATUS

approved

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Last modified February 20 04:39 EST 2018. Contains 299358 sequences. (Running on oeis4.)