A213582 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
1, 5, 2, 16, 9, 3, 42, 27, 13, 4, 99, 68, 38, 17, 5, 219, 156, 94, 49, 21, 6, 466, 339, 213, 120, 60, 25, 7, 968, 713, 459, 270, 146, 71, 29, 8, 1981, 1470, 960, 579, 327, 172, 82, 33, 9, 4017, 2994, 1972, 1207, 699, 384, 198, 93, 37, 10, 8100, 6053, 4007, 2474, 1454, 819, 441, 224, 104, 41, 11
Offset: 1
Examples
Northwest corner (the array is read by falling antidiagonals): 1...5....16...42....99....219 2...9....27...68....156...339 3...13...38...94....213...459 4...17...49...120...270...579 5...21...60...146...327...699 6...25...71...172...384...819
Links
- Clark Kimberling, Antidiagonals n = 1..60, flattened
Programs
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GAP
Flat(List([1..12], n-> List([1..n], k-> 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 ))); # G. C. Greubel, Jul 08 2019
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Magma
[[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019
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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}]] (* A213582 *) r[n_]:= Table[T[n, k], {k, 40}] Table[T[n, n], {n, 1, 40}] (* A213583 *) s[n_]:= Sum[T[i, n+1-i], {i, 1, n}] Table[s[n], {n, 1, 50}] (* A156928 *) (* Second program *) Table[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *) -
PARI
t(n,k) = 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2; for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019
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Sage
[[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019
Formula
T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-2*x) *(1-x)^3.
T(n,k) = 2*(n+1)*(2^k - 1) - k*(k + 2*n + 3)/2. - G. C. Greubel, Jul 08 2019
Comments