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 A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals. 5
 1, 3, 4, 7, 11, 10, 14, 23, 26, 20, 25, 42, 51, 50, 35, 41, 70, 88, 94, 85, 56, 63, 109, 140, 156, 155, 133, 84, 92, 161, 210, 240, 250, 237, 196, 120, 129, 228, 301, 350, 375, 374, 343, 276, 165, 175, 312, 416, 490, 535, 550, 532, 476, 375, 220, 231, 415, 558, 664, 735, 771, 770, 728, 639, 495, 286 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array.  The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.) The formula for the integer T(n,k) has denominator 12.  The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509. LINKS G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened FORMULA T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12. EXAMPLE The natural number array A000027 starts with   1, 2,  4,  7, ...   3, 5,  8, 12, ...   6, 9, 13, 18, ...   ... T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:    1,  3,   7,  14,  25,  41    4, 11,  23,  42,  70, 109   10, 26,  51,  88, 140, 210   20, 50,  94, 156, 240, 350   35, 85, 155, 250, 375, 535 MATHEMATICA f[n_, k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12; TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten CROSSREFS Cf. A000027, A185507, A185508, A185509. Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal). Sequence in context: A061447 A219188 A283428 * A328379 A080591 A291063 Adjacent sequences:  A185503 A185504 A185505 * A185507 A185508 A185509 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jan 29 2011 STATUS approved

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Last modified April 23 12:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)