A195311 -id:A195311 - cp's OEIS frontend

A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

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

Offset: 1

Omar E. Pol, Sep 21 2011

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.

			Written as a triangle:
   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;
  11, 10,  7,  5,  0;
  12, 11,  8,  6,  1;
  13, 12,  9,  7,  2;
  14, 13, 10,  8,  3,  0;
.
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.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Row sums give A195311.

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

rows = 20;
a1318[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];
T[n_, k_] := n - a1318[k];
Table[DeleteCases[Table[T[n, k], {k, 1, n}], ?Negative], {n, 1, rows}] // Flatten (* _Jean-François Alcover, Sep 22 2018 *)

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

Name essentially suggested by Franklin T. Adams-Watters (see history), Sep 21 2011

A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

Original entry on oeis.org

1, 9, 11, 45, 39, 126, 94, 270, 185, 495, 321, 819, 511, 1260, 764, 1836, 1089, 2565, 1495, 3465, 1991, 4554, 2586, 5850, 3289, 7371, 4109, 9135, 5055, 11160, 6136, 13464, 7361, 16065, 8739, 18981, 10279, 22230, 11990, 25830, 13881

Offset: 1

Omar E. Pol, Sep 21 2011

nonn

The integers in same rows of the source triangle have a property related to Euler's Pentagonal Theorem.

Note that the column 1 of the mentioned triangle gives the positive terms of A001318 (see example).

			a(1) = 1
a(2) = 2+3+4 = 9
a(3) = 5+6 = 11
a(4) = 7+8+9+10+11 = 45
a(5) = 12+13+14 = 39
a(6) = 15+16+17+18+19+20+21 = 126
a(7) = 22+23+24+25 = 94
a(8) = 26+27+28+29+30+31+32+33+34 = 270
a(9) = 35+36+37+38+39 = 185

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A026741, A195310, A195311, A004188 (bisection).

A195309 := proc(n)
        (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32
end proc:
seq(A195309(n),n=1..60) ; # R. J. Mathar, Oct 08 2011

LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,9,11,45,39,126,94,270},80] (* Harvey P. Dale, Jun 22 2015 *)

a(n) = (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32 . - R. J. Mathar, Oct 08 2011

G.f. x*(1+9*x+7*x^2+9*x^3+x^4) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Oct 08 2011

cp's OEIS Frontend

A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

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