A213582 - OEIS

%I #12 Sep 08 2022 08:46:02

%S 1,5,2,16,9,3,42,27,13,4,99,68,38,17,5,219,156,94,49,21,6,466,339,213,

%T 120,60,25,7,968,713,459,270,146,71,29,8,1981,1470,960,579,327,172,82,

%U 33,9,4017,2994,1972,1207,699,384,198,93,37,10,8100,6053,4007,2474,1454,819,441,224,104,41,11

%N Rectangular array:  (row n) = b**c, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213583.

%C Antidiagonal sums: A156928.

%C Row 1, (1,3,7,15,31,...)**(1,2,3,4,5,...): A002662.

%C Row 2, (1,3,7,15,31,...)**(2,3,4,5,6,...)

%C Row 3, (1,3,7,15,31,...)**(3,4,5,6,7,...)

%C For a guide to related arrays, see A213500.

%H Clark Kimberling, <a href="/A213582/b213582.txt">Antidiagonals n = 1..60, flattened</a>

%F T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).

%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)^3.

%F T(n,k) = 2*(n+1)*(2^k - 1) - k*(k + 2*n + 3)/2. - _G. C. Greubel_, Jul 08 2019

%e Northwest corner (the array is read by falling antidiagonals):

%e 1...5....16...42....99....219

%e 2...9....27...68....156...339

%e 3...13...38...94....213...459

%e 4...17...49...120...270...579

%e 5...21...60...146...327...699

%e 6...25...71...172...384...819

%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}]] (* A213582 *)

%t r[n_]:= Table[T[n, k], {k, 40}]

%t Table[T[n, n], {n, 1, 40}] (* A213583 *)

%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]

%t Table[s[n], {n, 1, 50}] (* A156928 *)

%t (* Second program *)

%t 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 *)

%o (PARI) t(n,k) = 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2;

%o for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 08 2019

%o (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

%o (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

%o (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

%Y Cf. A213500, A213571.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 19 2012