cp's OEIS Frontend

A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.

%I A108044 #52 Aug 22 2025 11:03:04
%S A108044 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,
%T A108044 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,
%U A108044 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
%N A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.
%C A108044 Column k has e.g.f. Bessel_I(k,2x). - _Paul Barry_, Mar 10 2010
%C A108044 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
%C A108044 T(n,k) is the number of walks on the semi-infinite 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
%H A108044 Harvey P. Dale, <a href="/A108044/b108044.txt">Rows n = 0..125 of triangle, flattened</a>
%H A108044 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, section 8.
%H A108044 Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 14.
%H A108044 Robert S. Maier, <a href="https://arxiv.org/abs/2508.13094">Sheffer Polynomials and the s-ordering of Exponential Boson Operators</a>, arXiv:2508.13094 [quant-ph], 2025. See p. 27.
%H A108044 Paul Peart and Wen-Jin Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%H A108044 Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.
%H A108044 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A108044 Each entry is the sum of those in the previous row that are to its left and to its right.
%F A108044 Riordan array (1/sqrt(1-4*x^2), (1-sqrt(1-4*x^2))/(2*x)).
%F A108044 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
%F A108044 From _Peter Bala_, Jun 29 2015: (Start)
%F A108044 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).
%F A108044 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) ).
%F A108044 The inverse array is A108045 (a hitting time array with h(x) = x/(1 + x^2)). (End)
%e A108044 Triangle begins:
%e A108044   1
%e A108044   0 1
%e A108044   2 0 1
%e A108044   0 3 0 1
%e A108044   6 0 4 0 1
%e A108044   0 10 0 5 0 1
%e A108044   20 0 15 0 6 0 1
%e A108044 From _Paul Barry_, Mar 10 2010: (Start)
%e A108044 Production matrix is
%e A108044   0, 1,
%e A108044   2, 0, 1,
%e A108044   0, 1, 0, 1,
%e A108044   0, 0, 1, 0, 1,
%e A108044   0, 0, 0, 1, 0, 1,
%e A108044   0, 0, 0, 0, 1, 0, 1,
%e A108044   0, 0, 0, 0, 0, 1, 0, 1,
%e A108044   0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A108044   0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
%e A108044 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
%p A108044 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
%t A108044 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 *)
%o A108044 (Haskell)
%o A108044 import Data.List (intersperse)
%o A108044 a108044 n k = a108044_tabl !! n !! k
%o A108044 a108044_row n = a108044_tabl !! n
%o A108044 a108044_tabl = zipWith drop [0..] $ map (intersperse 0) a007318_tabl
%o A108044 -- _Reinhard Zumkeller_, May 18 2013
%Y A108044 Cf. A007318, A108045 (matrix inverse),
%Y A108044 Cf. A204293, A357136, A000984 (column 0), A001700 (column 1), A001791 (column 2), A002054 (column 3)
%K A108044 nonn,tabl,easy,changed
%O A108044 0,4
%A A108044 _N. J. A. Sloane_, Jun 02 2005
%E A108044 More terms from _Emeric Deutsch_ and _Christian G. Bower_, Jun 05 2005