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%I #25 Nov 10 2024 02:22:07
%S 1,1,1,1,2,1,1,3,4,1,1,4,7,7,1,1,5,10,13,11,1,1,6,13,19,21,16,1,1,7,
%T 16,25,31,31,22,1,1,8,19,31,41,46,43,29,1,1,9,22,37,51,61,64,57,37,1,
%U 1,10,25,43,61,76,85,85,73,46,1,1,11,28,49,71,91,106,113,109,91
%N Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.
%C T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
%C It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
%C The triangle under the main diagonal is A121722.
%C Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - _Gary W. Adamson_, Oct 01 2015
%H Kival Ngaokrajang, <a href="/A244911/a244911.pdf">Illustration for n = 0..3, k = 1..4</a>
%F T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.
%e Table begins:
%e C0 C1 C2 C3 C4 C5
%e n/k 0 1 2 3 4 5 ...
%e R0 0 1 1 1 1 1 1 ...
%e R1 1 1 2 3 4 5 6 ...
%e R2 2 1 4 7 10 13 16 ...
%e R3 3 1 7 13 19 25 31 ...
%e R4 4 1 11 21 31 41 51 ...
%e R5 5 1 16 31 46 61 76 ...
%e R6 6 1 22 43 64 85 106 ...
%e R7 7 1 29 57 85 113 141 ...
%e R8 8 1 37 73 109 145 181 ...
%e R9 9 1 46 91 136 181 226 ...
%e ... ... ... ... ... ... ... ...
%e C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12 = A003154.
%e R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.
%o (Small Basic)
%o For k = 0 to 50
%o a[0][k] = 1
%o For n = 1 to 50
%o a[n][k] = n*k + a[n-1][k]
%o EndFor
%o Endfor
%o '==================================
%o For t = 1 to 20
%o d = 1
%o For nn = 0 To t-1
%o kk = t- d
%o TextWindow.Write(a[nn][kk]+", ")
%o d = d + 1
%o EndFor
%o Endfor
%Y Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.
%K nonn,tabl
%O 0,5
%A _Kival Ngaokrajang_, Jul 07 2014