A260492 Pascal's triangle aerated with columns of zeros.

1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1, 1, 0, 7, 0, 21, 0, 35, 0, 35, 0, 21, 0, 7, 0, 1, 1, 0, 8, 0, 28, 0, 56, 0, 70, 0, 56, 0, 28, 0, 8, 0, 1, 1, 0, 9, 0, 36, 0, 84, 0, 126, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 1, 0, 10, 0, 45, 0, 120, 0, 210, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1

Offset

0,7

Comments

To obtain this array we take Pascal's triangle A007318 and insert a column of zeros at columns 1, 3, 5, .... Thus the n-th row of this array gives the coefficients in the expansion of (1 + x^2)^n.

The transpose of this array is a generalized Riordan array (1, 1 + x^2) as defined by Wilson in 2005 (see the Bala link for details). Note that these are not the same as the generalized Riordan arrays introduced by Wang and Wang in 2008.

Call this array C and let p(x) = 1 + x^2. Then C^2 gives the coefficients in the expansion of the polynomials ( p(p(x)) )^n, C^3 gives the coefficients in the expansion of the polynomials ( p(p(p(x))) )^n and so on.

See A204293 for Pascal's triangle aerated by both row and columns.

The triangle is read by rows of lengths equal to the odd numbers 2*n + 1 = A005408(n), n >= 0; although it is then (from the 3rd line of comments & formula on) considered as an infinite square matrix C, the upper right being filled with zeros. - M. F. Hasler, Aug 19 2015

Links

Table of n, a(n) for n=0..120.

P. Bala, Notes on generalized Riordan arrays

Evgeniy Burlachenko, Riordan arrays and generalized Euler polynomials, arXiv:1709.02229 [math.NT], 2017.

Evgeniy Burlachenko, Riordan arrays, Chebyshev polynomials, Fibonacci bases, arXiv:1903.10435 [math.NT], 2019.

Alan D. Sokal, How to generalize (and not to generalize) the Chu-Vandermonde identity, arXiv:1804.08919 [math.CO], 2018.

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Formula

T(n,k) = (1 - k mod 2) binomial(n,floor(k/2)).

O.g.f.: 1/(1 - (1 + x^2)*t) = 1 + (1 + x^2)*t + (1 + 2*x^2 + x^4)*t^2 + ....

Let C denote this array.

Row sums of C = 2^n; Row sums of C^2 = 5^n; Row sums of C^3 = 26^n; Row of sums C^4 = 677^n. In general the row sums of C^m = A003095(m)^n.

First column of C^2 = 2^n; first column of C^3 = 5^n, first column of C^4 = 26^n and so on.

Let P denote Pascal's triangle A007318. Then C * transpose(C) = P * transpose(P) = the square symmetric Pascal matrix.

For n >= 0, (P^n)*C is the array P^(n+1) aerated by columns.

First column of P*C = 2^n; first column of (P*C)^2 = 6^n; first column of (P*C)^3 = 38^n, and so on, where [2, 6, 38, ...] is A072191.

Let R equal Pascal's triangle aerated with rows of zeros. Then C*R = P^2.

R*P*C is P^3 aerated by both rows and columns

Conjecturally, the limit of R^n * C^n as n -> oo has as its first column an aerated version of A027826, with zeros elsewhere in the array.

Aeration by rows/columns amounts to multiplication to the left/right by the identity matrix aerated by rows/column: R = J*P, C = P*J'. This makes obvious that P^n*C = P^(n+1)*J', R*P^n = J*P^(n+1), R*P^n*C = J*P^(n+2)*J' (aerated by both rows and columns). - M. F. Hasler, Aug 19 2015

Example

Table begins
n\k 0  1  2  3  4  5  6  7  8  9 10 11 12
- - - - - - - - - - - - - - - - - - - - -
0   1
1   1  0  1
2   1  0  2  0  1
3   1  0  3  0  3  0  1
4   1  0  4  0  6  0  4  0  1
5   1  0  5  0 10  0 10  0  5  0  1
6   1  0  6  0 15  0 20  0 15  0  6  0  1
...

Maple

#A260492
#define the aerated Pascal matrix (with indexing starting at 1)
C := Matrix(30, 30, (i, j) -> (1 - mod(j-1, 2))*binomial(i-1, floor((j-1)/2))):
for n from 1 to 12 do seq(C(n, k), k = 1 .. 2*n-1) end do;
# alternative program using row polynomials
rowpoly:= proc(n) option remember; expand((1 + x^2)^n) end:
T := (n, k)-> coeff(rowpoly(n), x, k):
seq(seq(T(n, k), k = 0..2*n), n = 0..11);

Mathematica

T[n_, k_] := If[OddQ[k], 0, Binomial[n, k/2]];
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Mar 23 2018 *)

Prog

(PARI) C=matrix(20, 20, m, n, if(n%2, binomial(m-1, n\2))); a=concat(vector(5, i, vector(i*2-1, j, C[i, j]))) \\ (C is read by rows, not antidiagonals.) - M. F. Hasler, Aug 19 2015

Crossrefs

Cf. A003095, A027826, A007318, A034870, A072191, A161869, A204293.

Sequence in context: A301568 A036851 A036850 * A308400 A369816 A236541

Adjacent sequences: A260489 A260490 A260491 * A260493 A260494 A260495

Keyword

nonn,tabf,easy

Author

Peter Bala, Aug 15 2015

Status

approved