A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
1, 5, 3, 16, 13, 7, 42, 38, 29, 15, 99, 94, 82, 61, 31, 219, 213, 198, 170, 125, 63, 466, 459, 441, 406, 346, 253, 127, 968, 960, 939, 897, 822, 698, 509, 255, 1981, 1972, 1948, 1899, 1809, 1654, 1402, 1021, 511, 4017, 4007, 3980, 3924, 3819, 3633
Offset: 1
Examples
Northwest corner (the array is read by falling antidiagonals): 1, 5, 16, 42, 99, 219, ... 3, 13, 38, 94, 213, 459, ... 7, 29, 82, 198, 441, 939, ... 15, 61, 170, 406, 897, 1899, ... 31, 125, 346, 822, 1809, 3819, ... ...
Links
- Clark Kimberling, antidiagonals n = 1..60, flattened
Programs
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GAP
Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # G. C. Greubel, Jul 25 2019
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Magma
[2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019
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Mathematica
(* First program *) b[n_]:= n; c[n_]:= -1 + 2^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}] (* A213571 *) d = Table[t[n, n], {n, 1, 40}] (* A213572 *) s[n_]:= Sum[t[i, n+1-i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A213581 *) (* Additional programs *) Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n,12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *) -
PARI
for(n=1,12, for(k=1,n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Jul 25 2019
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Sage
[[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 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) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.
T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - G. C. Greubel, Jul 25 2019
Comments