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 A157638 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 2. 3
 1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 60, 210, 60, 1, 1, 155, 1550, 1550, 155, 1, 1, 378, 9765, 27900, 9765, 378, 1, 1, 889, 56007, 413385, 413385, 56007, 889, 1, 1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1, 1, 4599, 1563660, 66194940 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Other triangles in the family (see name) include: q = 2 (this triangle), q = 3 (see A157640), and q = 4 (see A157641). - Werner Schulte, Nov 16 2018 LINKS Andrew Howroyd, Rows n=0..49 of triangle, flattened FORMULA T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*2^i. T(n,k) = binomial(n,k)*A022166(n,k) for 0 <= k <= n. - Werner Schulte, Nov 14 2018 T(n,k) = n!*A005329(n)/(k!*A005329(k)*(n-k)!*A005329(n-k)). - Andrew Howroyd, Nov 14 2018 EXAMPLE Triangle begins:   1;   1, 1;   1, 6, 1;   1, 21, 21, 1;   1, 60, 210, 60, 1;   1, 155, 1550, 1550, 155, 1;   1, 378, 9765, 27900, 9765, 378, 1;   1, 889, 56007, 413385, 413385, 56007, 889, 1;   1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1;   1, 4599, 1563660, 66194940, 417028122, 417028122, 66194940, 1563660, 4599, 1; MATHEMATICA t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]; b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]); Flatten[Table[Table[b[n, k, 1], {k, 0, n}], {n, 0, 10}]] PROG (PARI) T(n, k) = {binomial(n, k)*polcoef(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)} \\ Andrew Howroyd, Nov 14 2018 (PARI) q=2; for(n=0, 10, for(k=0, n, print1(binomial(n, k)*prod(j=0, k-1, (1-q^(n-j))/(1-q^(j+1))), ", "))) \\ G. C. Greubel, Nov 17 2018 (MAGMA) q:=2; [[k le 0 select 1 else Binomial(n, k)*(&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 17 2018 (Sage) [[ binomial(n, k)*gaussian_binomial(n, k).subs(q=2) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 17 2018 CROSSREFS Cf. A007318, A005329, A022166, A157640, A157641. Sequence in context: A144066 A296827 A056941 * A142596 A176063 A155467 Adjacent sequences:  A157635 A157636 A157637 * A157639 A157640 A157641 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Mar 03 2009 EXTENSIONS Edited and simpler name by Werner Schulte and Andrew Howroyd, Nov 14 2018 STATUS approved

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Last modified January 18 00:56 EST 2021. Contains 340249 sequences. (Running on oeis4.)