cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-13 of 13 results.

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

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

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.

Examples

			.  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;

Crossrefs

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

A195841 Triangle read by rows which arises from A195831, in the same way as A175003 arises from A195310. Column k starts at row A195160(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized hendecagonal numbers A195160, A195831 and A195851 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Crossrefs

Row sums give A195851.
Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840.

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

Links

Crossrefs

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.