|
%I #22 Sep 04 2023 12:45:49
%S 1,4,2,11,7,3,26,18,10,4,57,41,25,13,5,120,88,56,32,16,6,247,183,119,
%T 71,39,19,7,502,374,246,150,86,46,22,8,1013,757,501,309,181,101,53,25,
%U 9,2036,1524,1012,628,372,212,116,60,28,10,4083,3059,2035,1267
%N Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213569
%C Antidiagonal sums: A047520
%C Row 1, (1,3,6,...)**(1,4,9,...): A125128
%C Row 2, (1,3,6,...)**(4,9,16,...): A095151
%C Row 3, (1,3,6,...)**(9,16,25,...): A000247
%C Row 4, (1,3,6,...)**(16,25,36...): A208638 (?)
%C For a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213568/b213568.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by _Clark Kimberling_, Sep 03 2023
%F 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.
%F T(n,k) = 2^k*(n + 1) - (n + k + 1). - _G. C. Greubel_, Jul 26 2019
%e Northwest corner (the array is read by falling antidiagonals):
%e 1...4....11...26....57....120
%e 2...7....18...41....88....183
%e 3...10...25...56....119...246
%e 4...13...32...71....150...309
%e 5...16...39...86....181...372
%e 6...19...46...101...212...435
%t (* First program *)
%t b[n_]:= 2^(n-1); c[n_]:= n;
%t t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t r[n_]:= Table[t[n, k], {k, 1, 60}] (* A213568 *)
%t d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
%t s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
%t (* Second program *)
%t Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 26 2019 *)
%o (PARI) for(n=1,12, for(k=1,n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ _G. C. Greubel_, Jul 26 2019
%o (Magma) [2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 26 2019
%o (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
%o (GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # _G. C. Greubel_, Jul 26 2019
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 18 2012
|
