A195310 - OEIS

%I #63 Sep 23 2018 22:28:29

%S 0,1,0,2,1,3,2,4,3,0,5,4,1,6,5,2,0,7,6,3,1,8,7,4,2,9,8,5,3,10,9,6,4,

%T 11,10,7,5,0,12,11,8,6,1,13,12,9,7,2,14,13,10,8,3,0,15,14,11,9,4,1,16,

%U 15,12,10,5,2,17,16,13,11,6,3,18,17,14,12,7,4

%N Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

%C Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A001318(k). This sequence is related to Euler's Pentagonal Number Theorem. A000041(a(n)) gives the absolute value of A175003(n). To get the number of partitions of n see the example.

%H L. Euler, <a href="http://www.math.dartmouth.edu/~euler/docs/originals/E542.pdf">De mirabilibus proprietatibus numerorum pentagonalium</a>

%H L. Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>

%F A175003(n,k) = A057077(k-1)*A000041(T(n,k)), n >= 1, k >= 1.

%e Written as a triangle:

%e    0;

%e    1,  0;

%e    2,  1;

%e    3,  2;

%e    4,  3,  0;

%e    5,  4,  1;

%e    6,  5,  2,  0;

%e    7,  6,  3,  1;

%e    8,  7,  4,  2;

%e    9,  8,  5,  3;

%e   10,  9,  6,  4;

%e   11, 10,  7,  5,  0;

%e   12, 11,  8,  6,  1;

%e   13, 12,  9,  7,  2;

%e   14, 13, 10,  8,  3,  0;

%e .

%e For n = 15, consider row 15 which lists the numbers 14, 13, 10, 8, 3, 0. From Euler's Pentagonal Number Theorem we have that the number of partitions of 15 is p(15) = p(14) + p(13) - p(10) - p(8) + p(3) + p(0) = 135 + 101 - 42 - 22 + 3 + 1 = 176.

%t rows = 20;

%t a1318[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];

%t T[n_, k_] := n - a1318[k];

%t Table[DeleteCases[Table[T[n, k], {k, 1, n}], _?Negative], {n, 1, rows}] // Flatten (* _Jean-François Alcover_, Sep 22 2018 *)

%Y Row sums give A195311.

%Y Cf. A000041, A001318, A010815, A026741, A057077, A175003.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Sep 21 2011

%E Name essentially suggested by _Franklin T. Adams-Watters_ (see history), Sep 21 2011