A195825 -id:A195825 - cp's OEIS frontend

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

1, 1, 2, 4, 6, 10, 16, 24, 36, 54, 78, 112, 160, 224, 312, 432, 590, 802, 1084, 1452, 1936, 2568, 3384, 4440, 5800, 7538, 9758, 12584, 16160, 20680, 26376, 33520, 42468, 53644, 67552, 84832, 106246, 132706, 165344, 205512, 254824, 315256, 389168, 479368

Offset: 0

Omar E. Pol, Jun 10 2012

nonn

Partial sums give A015128. - Omar E. Pol, Jan 09 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A006950, A008794, A036820, A057077, A195152, A195848, A195849, A195850, A195851, A195852, A196933, A195825, A210964, A277643.

Flatten[{1, Differences[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 60}]]}] (* Vaclav Kotesovec, Oct 25 2016 *)
CoefficientList[Series[(1 - x)/EllipticTheta[4, 0, x], {x, 0, 43}], x] (* Robert G. Wilson v, Mar 06 2018 *)

a(n) ~ exp(Pi*sqrt(n))*Pi / (16*n^(3/2)) * (1 - (3/Pi + Pi/4)/sqrt(n) + (3/2 + 3/Pi^2+ Pi^2/24)/n). - Vaclav Kotesovec, Oct 25 2016, extended Nov 04 2016

G.f.: (1 - x)/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Mar 05 2018

A195828 Triangle read by rows with T(n,k) = n - A001082(k), n>=1, k>=1, if (n - A001082(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 5, 1, 6, 2, 7, 3, 0, 8, 4, 1, 9, 5, 2, 10, 6, 3, 11, 7, 4, 12, 8, 5, 13, 9, 6, 14, 10, 7, 15, 11, 8, 0, 16, 12, 9, 1, 17, 13, 10, 2, 18, 14, 11, 3, 19, 15, 12, 4, 20, 16, 13, 5, 0, 21, 17, 14, 6, 1, 22, 18, 15, 7, 2, 23, 19, 16, 8, 3

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A001082(k).

This sequence is related to the generalized octagonal numbers A001082, A195838 and A195848 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4,  0;
.  5,  1;
.  6,  2;
.  7,  3,  0;
.  8,  4,  1;
.  9,  5,  2;
. 10,  6,  3;
. 11,  7,  4;
. 12,  8,  5;
. 13,  9,  6;
. 14, 10,  7;
. 15, 11,  8,  0;
. 16, 12,  9,  1;
. 17, 13, 10,  2;
. 18, 14, 11,  3;
. 19, 15, 12,  4;
. 20, 16, 13,  5;  0;
. 21, 17, 14,  6;  1;
. 22, 18, 15,  7;  2;

Cf. A001082, A195310, A195825, A195826, A195827, A195829, A195830, A195831, A195832, A195833, A195838, A195848.

A195836 Triangle read by rows which arises from A195826 in the same way as A175003 arises from A195310. Column k starts at row A000217(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 5, -2, 13, 7, -3, -1, 16, 10, -4, -1, 21, 13, -5, -1, 28, 16, -7, -2, 35, 21, -10, -3, 43, 28, -13, -4, 1, 55, 35, -16, -5, 1, 70, 43, -21, -7, 1, 86, 55, -28, -10, 2, 105, 70, -35, -13, 3, 130, 86, -43, -16, 4

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. It appears that this sequence is related to the generalized hexagonal numbers (A000217), A195826 and A006950 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. It appears that row sums give A006950. See comments in A195825.

			Written as a triangle:
.  1,
.  1,
.  1,   1,
.  2,   1,
.  3,   1,
.  4,   2,  -1,
.  5,   3,  -1,
.  7,   4,  -1,
. 10,   5,  -2,
. 13,   7,  -3,   -1,
. 16,  10,  -4,   -1,
. 21,  13,  -5,   -1,
. 28,  16,  -7,   -2,
. 35,  21,  -10,  -3,
. 43,  28,  -13,  -4,   1,
. 55,  35,  -16,  -5,   1,
. 70,  43,  -21,  -7,   1,
. 86,  55,  -28, -10,   2,

Cf. A000217, A006950, A175003, A195825, A195826, A195837, A195838, A195939, A195840, A195841, A195842, A195843.

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

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

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

A195826 Triangle read by rows with T(n,k) = n - A000217(k), n>=1, k>=1, if (n - A000217(k))>=0.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 5, 3, 0, 6, 4, 1, 7, 5, 2, 8, 6, 3, 9, 7, 4, 0, 10, 8, 5, 1, 11, 9, 6, 2, 12, 10, 7, 3, 13, 11, 8, 4, 14, 12, 9, 5, 0, 15, 13, 10, 6, 1, 16, 14, 11, 7, 2, 17, 15, 12, 8, 3, 18, 16, 13, 9, 4, 19, 17, 14, 10, 5, 20, 18, 15, 11, 6, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A000217(k).

This sequence is related to the generalized hexagonal numbers (A000217), A195836 and A006950 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2,  0;
.  3,  1;
.  4,  2;
.  5,  3,  0;
.  6,  4,  1;
.  7,  5,  2;
.  8,  6,  3;
.  9,  7,  4,  0;
. 10,  8,  5,  1;
. 11,  9,  6,  2;
. 12, 10,  7,  3;
. 13, 11,  8,  4;
. 14, 12,  9,  5,  0;
. 15, 13,  10, 6,  1;
. 16, 14,  11, 7,  2;
. 17, 15,  12, 8,  3;

Cf. A000217, A006950, A195310, A195825, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195836.

A195829 Triangle read by rows with T(n,k) = n - A118277(k), n>=1, k>=1, if (n - A118277(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 6, 1, 7, 2, 8, 3, 0, 9, 4, 1, 10, 5, 2, 11, 6, 3, 12, 7, 4, 13, 8, 5, 14, 9, 6, 15, 10, 7, 16, 11, 8, 17, 12, 9, 18, 13, 10, 0, 19, 14, 11, 1, 20, 15, 12, 2, 21, 16, 13, 3, 22, 17, 14, 4, 23, 18, 15, 5, 0, 24, 19, 16, 6, 1, 25, 20, 17, 7, 2

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A118277(k).

This sequence is related to the generalized enneagonal numbers A118277, A195839 and A195849 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5,  0;
.  6,  1;
.  7,  2;
.  8,  3,  0;
.  9,  4,  1;
. 10,  5,  2;
. 11,  6,  3;
. 12,  7,  4;
. 13,  8,  5;
. 14,  9,  6;
. 15, 10,  7;
. 16, 11,  8;
. 17, 12,  9;
. 18, 13, 10,  0;
. 19, 14, 11,  1;
. 20, 15, 12,  2;

Cf. A118277, A195310, A195825, A195826, A195827, A195828, A195830, A195831, A195832, A195833, A195839, A195849.

A195830 Triangle read by rows with T(n,k) = n - A074377(k), n>=1, k>=1, if (n - A074377(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 7, 1, 8, 2, 9, 3, 0, 10, 4, 1, 11, 5, 2, 12, 6, 3, 13, 7, 4, 14, 8, 5, 15, 9, 6, 16, 10, 7, 17, 11, 8, 18, 12, 9, 19, 13, 10, 20, 14, 11, 21, 15, 12, 0, 22, 16, 13, 1, 23, 17, 14, 2, 24, 18, 15, 3, 25, 19, 16, 4, 26, 20, 17, 5, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A074377(k).

This sequence is related to the generalized decagonal numbers A074377, A195840 and A195850 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6,  0;
.  7,  1;
.  8,  2;
.  9,  3,  0;
. 10,  4,  1;
. 11,  5,  2;
. 12,  6,  3;
. 13,  7,  4;
. 14,  8,  5;
. 15,  9,  6;
. 16, 10,  7;
. 17, 11,  8;
. 18, 12,  9;
. 19, 13, 10;
. 20, 14, 11;
. 21, 15, 12,  0;
. 22, 16, 13,  1;
. 23, 17, 14,  2;

Cf. A074377, A195310, A195825, A195826, A195827, A195828, A195829, A195831, A195832, A195833, A195840, A195850.

A195831 Triangle read by rows with T(n,k) = n - A195160(k), n>=1, k>=1, if (n - A195160(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 0, 8, 1, 9, 2, 10, 3, 0, 11, 4, 1, 12, 5, 2, 13, 6, 3, 14, 7, 4, 15, 8, 5, 16, 9, 6, 17, 10, 7, 18, 11, 8, 19, 12, 9, 20, 13, 10, 21, 14, 11, 22, 15, 12, 23, 16, 13, 24, 17, 14, 0, 25, 18, 15, 1, 26, 19, 16, 2, 27, 20, 17, 3, 28, 21, 18, 4

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195160(k).

This sequence is related to the generalized hendecagonal numbers A195160, A195841 and A195851 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7,  0;
.  8,  1;
.  9,  2;
. 10,  3,  0;
. 11,  4,  1;
. 12,  5,  2;
. 13,  6,  3;
. 14,  7,  4;
. 15,  8,  5;
. 16,  9,  6;
. 17, 10,  7;
. 18, 11,  8;
. 19, 12,  9;
. 20, 13, 10;

Cf. A195160, A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195832, A195833, A195841, A195851.

A195832 Triangle read by rows with T(n,k) = n - A195162(k), n>=1, k>=1, if (n - A195162(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 9, 1, 10, 2, 11, 3, 0, 12, 4, 1, 13, 5, 2, 14, 6, 3, 15, 7, 4, 16, 8, 5, 17, 9, 6, 18, 10, 7, 19, 11, 8, 20, 12, 9, 21, 13, 10, 22, 14, 11, 23, 15, 12, 24, 16, 13, 25, 17, 14, 26, 18, 15, 27, 19, 16, 0, 28, 20, 17, 1, 29, 21, 18, 2

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195162(k).

This sequence is related to the generalized dodecagonal numbers A195162, A195842 and A195852 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7;
.  8,  0;
.  9,  1;
. 10,  2;
. 11,  3,  0;
. 12,  4,  1;
. 13,  5,  2;
. 14,  6,  3;
. 15,  7,  4;
. 16,  8,  5;
. 17,  9,  6;
. 18, 10,  7;

Cf. A195162, A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195833, A195842, A195852.

A195833 Triangle read by rows with T(n,k) = n - A195313(k), n>=1, k>=1, if (n - A195313(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 1, 11, 2, 12, 3, 0, 13, 4, 1, 14, 5, 2, 15, 6, 3, 16, 7, 4, 17, 8, 5, 18, 9, 6, 19, 10, 7, 20, 11, 8, 21, 12, 9, 22, 13, 10, 23, 14, 11, 24, 15, 12, 25, 16, 13, 26, 17, 14, 27, 18, 15, 28, 19, 16, 29, 20, 17, 30, 21, 18, 0

Offset: 1

Omar E. Pol, Sep 24 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195313(k).

This sequence is related to the generalized tridecagonal numbers A195313, A195843 and A196933 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5;
.  6;
.  7;
.  8;
.  9, 0;
. 10, 1;
. 11, 2;
. 12, 3, 0;
. 13, 4, 1;
. 14, 5, 2;
. 15, 6, 3;
. 16, 7, 4;

Cf. A195310, A195313, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195843, A196933.

cp's OEIS Frontend

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

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A195828 Triangle read by rows with T(n,k) = n - A001082(k), n>=1, k>=1, if (n - A001082(k))>=0.

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A195836 Triangle read by rows which arises from A195826 in the same way as A175003 arises from A195310. Column k starts at row A000217(k).

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A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

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A195826 Triangle read by rows with T(n,k) = n - A000217(k), n>=1, k>=1, if (n - A000217(k))>=0.

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A195829 Triangle read by rows with T(n,k) = n - A118277(k), n>=1, k>=1, if (n - A118277(k))>=0.

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A195830 Triangle read by rows with T(n,k) = n - A074377(k), n>=1, k>=1, if (n - A074377(k))>=0.

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A195831 Triangle read by rows with T(n,k) = n - A195160(k), n>=1, k>=1, if (n - A195160(k))>=0.

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A195832 Triangle read by rows with T(n,k) = n - A195162(k), n>=1, k>=1, if (n - A195162(k))>=0.

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A195833 Triangle read by rows with T(n,k) = n - A195313(k), n>=1, k>=1, if (n - A195313(k))>=0.

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