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 A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k). 1
 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 8, 16, 13, 5, 1, 1, 10, 28, 32, 19, 6, 1, 1, 12, 44, 68, 55, 26, 7, 1, 1, 14, 64, 128, 136, 86, 34, 8, 1, 1, 16, 88, 220, 296, 241, 126, 43, 9, 1, 1, 18, 116, 352, 584, 592, 393, 176, 53, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From Emanuele Munarini, Feb 21 2017: (Start) T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps X=(1,0), D=(1,1) and E=(3,1). Row sums = A008998. Central coefficients = A006139. (End) LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened) James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018. FORMULA G.f.: 1/(1-y-x*(1+y^2)). From Emanuele Munarini, Feb 21 2017: (Start) G.f. for the triangle: 1/(1-x-x*y-x^3*y). Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n,k). (End) EXAMPLE 1; 1, 1; 1, 2, 1; 1, 4, 3, 1; 1, 6, 8, 4, 1; 1, 8, 16, 13, 5, 1; MATHEMATICA Table[Sum[Binomial[k, m] Binomial[n - 2 m, k], {m, 0, (n - k)/2}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 21 2015 *) PROG (Maxima) T(n, k):=sum(binomial(k, m)*binomial(n-2*m, k), m, 0, (n-k)/2); CROSSREFS Cf. A006139. Sequence in context: A130523 A034363 A026769 * A230858 A060098 A161492 Adjacent sequences:  A257362 A257363 A257364 * A257366 A257367 A257368 KEYWORD nonn,tabl AUTHOR Vladimir Kruchinin, Apr 21 2015 STATUS approved

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Last modified May 15 03:16 EDT 2021. Contains 343909 sequences. (Running on oeis4.)