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 A156197 T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows. 0
 2, 2, 2, 3, 4, 3, 6, 8, 8, 6, 15, 18, 18, 18, 15, 43, 47, 42, 42, 47, 43, 133, 138, 110, 96, 110, 138, 133, 430, 436, 324, 240, 240, 324, 436, 430, 1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431, 4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Table of n, a(n) for n=0..54. L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388. FORMULA T(n,k) = -binomial(k + n, -1 + k) + binomial(k + n, n) + binomial(-k + 2*n, n] - binomial(-k + 2*n, -1 - k + n). From Roger L. Bagula and Gary W. Adamson, Dec 03 2009: (Start) T(n,k) = ((n - k + 1)*binomial(n + k, n) + (k + 1)*binomial(-k + 2*n, n))/(n + 1). T(n,k) = A009766(n,k) + A033184(n,k). (End) G.f.: (C(t*x) + C(x)*(1 - x*C(t*x) - t*x*C(t*x)))/((1 - t*x*C(x))*(1 - x*C(t*x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x). - Franck Maminirina Ramaharo, Dec 11 2018 EXAMPLE Triangle begins: 2; 2, 2; 3, 4, 3; 6, 8, 8, 6; 15, 18, 18, 18, 15; 43, 47, 42, 42, 47, 43; 133, 138, 110, 96, 110, 138, 133; 430, 436, 324, 240, 240, 324, 436, 430; 1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431; 4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863; ... MATHEMATICA t0[n_, m_] = Binomial[n + m, n] - Binomial[n + m, m - 1]; T[n_, m_] = FullSimplify[t0[n, m] + t0[n, n - m]]; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}] // Flatten (* or *) Table[Table[((1 - k + n)*Binomial[k + n, n] + (1 + k)*Binomial[-k + 2*n, n])/(1 + n), {k, 0, n}], {n, 0, 10}] // Flatten (* Roger L. Bagula and Gary W. Adamson, Dec 03 2009 *) PROG (Maxima) A009766(n, k) := binomial(n + k, n)*(n - k + 1)/(n + 1)\$ create_list(A009766(n, k) + A009766(n, n - k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 11 2018 */ CROSSREFS Row sums: 2*A000108 (without the first term). Cf. A009766, A033184. - Roger L. Bagula and Gary W. Adamson, Dec 03 2009. Sequence in context: A098745 A029158 A241953 * A155162 A275444 A243322 Adjacent sequences: A156194 A156195 A156196 * A156198 A156199 A156200 KEYWORD nonn,tabl,easy AUTHOR Roger L. Bagula, Feb 05 2009 EXTENSIONS Edited by Franck Maminirina Ramaharo, Dec 11 2018 STATUS approved

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Last modified June 6 14:05 EDT 2023. Contains 363147 sequences. (Running on oeis4.)