

A108044


Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.


9



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
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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 (nk)/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
T(n,k) is the number of walks on the semiinfinite integer line with n steps that end at k. The walks start at 0, move at each step either one to the left or one to the right, and never enter the region of negative k. [Walks with impenetrable wall at 1/2. Dyck excursions of n steps that end at level k.] The variant without the restriction of negative positions is A053121.  R. J. Mathar, Nov 02 2023


LINKS

L. W. Shapiro, S. Getu, WenJin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229239.


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(14*x^2), (1sqrt(14*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/(1tg), where f=1/sqrt(14x^2) and g=(1sqrt(14x^2))/(2x).  Emeric Deutsch, Jun 05 2005
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^(nk)] 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^(nk)] 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
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


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



