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A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's. 8
1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 6, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 20, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 70, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 924, 0, 792, 0, 495, 0, 220, 0, 66 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column k has e.g.f. Bessel_I(k,2x). - Paul Barry, Mar 10 2010

T(n,k) is the number of binary sequences of length n in which the number of ones minus the number of zeros is k. If 2 divides(n+k), such a sequence will have (n+k)/2 ones and (n-k)/2 zeros. Since there are C(n,(n+k)/2) ways to choose the sequence entries that get a one, T(n,k)=binomial(n,(n+k)/2) whenever (n+k) is even and T(n,k)= 0 otherwise. See the example below in the example section. - Dennis P. Walsh, Apr 11 2012

LINKS

Harvey P. Dale, Rows n = 0..125 of triangle, flattened

P. Barry, A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 8.

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 14.

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

Each entry is the sum of those in the previous row that are to its left and to its right.

Riordan array (1/sqrt(1-4*x^2), (1-sqrt(1-4*x^2))/(2*x)).

T(n, k) = binomial(n, (n+k)/2) if n+k is even, T(n, k)=0 if n+k is odd. G.f.=f/(1-tg), where f=1/sqrt(1-4x^2) and g=(1-sqrt(1-4x^2))/(2x). - Emeric Deutsch, Jun 05 2005

From Peter Bala, Jun 29 2015: (Start)

Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - sqrt(1 - 4*x) )/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).

T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + x^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ).

The inverse array is A108045 (a hitting time array with h(x) = x/(1 + x^2)). (End)

EXAMPLE

Triangle begins:

.1

.0 1

.2 0 1

.0 3 0 1

.6 0 4 0 1

.0 10 0 5 0 1

.20 0 15 0 6 0 1

From Paul Barry, Mar 10 2010: (Start)

Production matrix is

0, 1,

2, 0, 1,

0, 1, 0, 1,

0, 0, 1, 0, 1,

0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)

T(5,1)=10 since there are 10 binary sequences of length 5 in which the number of ones exceed the number of zeros by exactly 1, namely, 00111, 01011, 01101, 01110, 10011, 10101, 10110, 11001, 11010, and 11100. Also, T(5,2)=0 since there are no binary sequences in which the number of ones exceed the number of zeros by exactly 2. - Dennis P. Walsh, Apr 11 2012

MAPLE

T:=proc(n, k) if n+k mod 2 = 0 then binomial(n, (n+k)/2) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Jun 05 2005

MATHEMATICA

b[n_, k_]:=If[EvenQ[n+k], Binomial[n, (n+k)/2], 0]; Flatten[Table[b[n, k], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, May 05 2013 *)

PROG

(Haskell)

import Data.List (intersperse)

a108044 n k = a108044_tabl !! n !! k

a108044_row n = a108044_tabl !! n

a108044_tabl = zipWith drop [0..] $ map (intersperse 0) a007318_tabl

-- Reinhard Zumkeller, May 18 2013

CROSSREFS

Cf. A007318, A108045 (matrix inverse).

Cf. A204293.

Sequence in context: A134511 A112554 A120616 * A104477 A224928 A052173

Adjacent sequences:  A108041 A108042 A108043 * A108045 A108046 A108047

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Jun 02 2005

EXTENSIONS

More terms from Emeric Deutsch and Christian G. Bower, Jun 05 2005

STATUS

approved

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Last modified February 21 17:43 EST 2018. Contains 299414 sequences. (Running on oeis4.)