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 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 (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 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 LINKS Harvey P. Dale, Rows n = 0..125 of triangle, flattened Paul Barry and 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, A357136, A000984 (column 0), A001700 (column 1), A001791 (column 2), A002054 (column 3) Sequence in context: A134511 A112554 A120616 * A104477 A224928 A308066 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 August 15 11:56 EDT 2024. Contains 375173 sequences. (Running on oeis4.)