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 A213568 Rectangular array:  (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 8
 1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal:  A213569 Antidiagonal sums:  A047520 Row 1,  (1,3,6,...)**(1,4,9,...):  A125128 Row 2,  (1,3,6,...)**(4,9,16,...):  A095151 Row 3,  (1,3,6,...)**(9,16,25,...):  A000247 Row 4,  (1,3,6,...)**(16,25,36...):  A208638 (?) For a guide to related arrays, see A213500. LINKS Clark Kimberling, Antidiagonals n = 1..60, flattened FORMULA T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-2). G.f. for row n:  f(x)/g(x), where f(x) = n - (n - 1)*x and g(x) = (1 - 2*x)*(1 - x)^2. T(n,k) = 2^k*(n + 1) - (n + k + 1). - G. C. Greubel, Jul 26 2019 EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1...4....11...26....57....120 2...7....18...41....88....183 3...10...25...56....119...246 4...13...32...71....150...309 5...16...39...86....181...372 6...19...46...101...212...435 MATHEMATICA (* First program *) b[n_]:= 2^(n-1); c[n_]:= n; t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}] TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]] Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *) d = Table[t[n, n], {n, 1, 40}] (* A213569 *) s[n_]:= Sum[t[i, n+1-i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A047520 *) (* Second program *) Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 26 2019 *) PROG (PARI) for(n=1, 12, for(k=1, n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ G. C. Greubel, Jul 26 2019 (MAGMA) [2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019 (Sage) [[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019 (GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # G. C. Greubel, Jul 26 2019 CROSSREFS Cf. A213500. Sequence in context: A163918 A318784 A154699 * A185878 A182870 A094406 Adjacent sequences:  A213565 A213566 A213567 * A213569 A213570 A213571 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jun 18 2012 STATUS approved

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Last modified May 25 17:53 EDT 2020. Contains 334595 sequences. (Running on oeis4.)