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A071675 Array read by antidiagonals of trinomial coefficients. 19
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 6, 16, 15, 6, 1, 0, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 0, 1, 30, 141, 266, 266 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = Sum_{i=0..floor((n+k)/2)} C(k,2i+2k-n)*C(2i+2k-n,i). Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

FORMULA

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.

As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - Paul Barry, Apr 26 2005

EXAMPLE

Rows start

1, 0,  0,  0,  0,  0, ...;

1, 1,  1,  0,  0,  0,  0, ...;

1, 2,  3,  2,  1,  0,  0, ...;

1, 3,  6,  7,  6,  3,  1, 0, ...;

1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.

MATHEMATICA

T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,

Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

CROSSREFS

Visible version of A027907. Row sums are 3^n, i.e. A000244. Central diagonal is A002426. Cf. A071676 for a slight variation.

Sequence in context: A173402 A055334 A106237 * A221833 A319082 A034365

Adjacent sequences:  A071672 A071673 A071674 * A071676 A071677 A071678

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, May 30 2002

STATUS

approved

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Last modified October 27 13:45 EDT 2021. Contains 348276 sequences. (Running on oeis4.)