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 A174124 Triangle T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 1, read by rows. 5
 1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 20, 40, 20, 1, 1, 30, 100, 100, 30, 1, 1, 42, 210, 350, 210, 42, 1, 1, 56, 392, 980, 980, 392, 56, 1, 1, 72, 672, 2352, 3528, 2352, 672, 72, 1, 1, 90, 1080, 5040, 10584, 10584, 5040, 1080, 90, 1, 1, 110, 1650, 9900, 27720, 38808, 27720, 9900, 1650, 110, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Triangles of this class, depending upon q, are of the form T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and have the row sums Sum_{k=0..n} T(n, k, q) = q*(q+1)*C_{n+q}/binomial(n+2*q, q-1) - 2*q + q*[n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. - G. C. Greubel, Feb 11 2021 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. FORMULA Let c(n, q) = Product_{j=2..n} j*(j+q) for n > 2, otherwise 1, then the number triangle is given by T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) for q = 1. From G. C. Greubel, Feb 11 2021: (Start) T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 1. Sum_{k=0..n} T(n, k, 1) = 2*A000108(n+1) - 2 + [n=0]. (End) EXAMPLE Triangle begins as:   1;   1,   1;   1,   6,    1;   1,  12,   12,    1;   1,  20,   40,   20,     1;   1,  30,  100,  100,    30,     1;   1,  42,  210,  350,   210,    42,     1;   1,  56,  392,  980,   980,   392,    56,    1;   1,  72,  672, 2352,  3528,  2352,   672,   72,    1;   1,  90, 1080, 5040, 10584, 10584,  5040, 1080,   90,   1;   1, 110, 1650, 9900, 27720, 38808, 27720, 9900, 1650, 110, 1; MATHEMATICA (* First program *) c[n_, q_]:= If[n<2, 1, Product[i*(i+q), {i, 2, n}]]; T[n_, m_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]); Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* Second program *) T[n_, k_, q_]:= If[k==0 || k==n, 1, (q+1)*Binomial[n, k]*(Pochhammer[q+1, n]/(Pochhammer[q+1, k]*Pochhammer[q+1, n-k]))]; Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2021 *) PROG (Sage) def T(n, k, q): return 1 if (k==0 or k==n) else (q+1)*binomial(n, k)*(rising_factorial(q+1, n)/(rising_factorial(q+1, k)*rising_factorial(q+1, n-k))) flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021 (Magma) c:= func< n, q | n lt 2 select 1 else (&*[j*(j+q): j in [2..n]]) >; T:= func< n, k, q | c(n, q)/(c(k, q)*c(n-k, q)) >; [T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021 CROSSREFS Cf. this sequence (q=1), A174125 (q=2). Cf. A000108, A174116, A174117, A174119. Sequence in context: A146772 A202868 A202877 * A174345 A174449 A174150 Adjacent sequences:  A174121 A174122 A174123 * A174125 A174126 A174127 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Mar 09 2010 EXTENSIONS Edited by G. C. Greubel, Feb 11 2021 STATUS approved

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Last modified September 17 22:14 EDT 2021. Contains 347489 sequences. (Running on oeis4.)