A195848 -id:A195848 - cp's OEIS frontend

A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 194, 197, 203, 216, 242, 281, 328, 367, 393, 407

Offset: 0

Omar E. Pol, Oct 07 2011

Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13], [35, 35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 9 (mod 10). - Peter Bala, Dec 10 2020

Cf. A000041, A001082, A006950, A036820, A057077, A195162, A195825, A195832, A195848, A195849, A195850, A195851, A196933, A210964, A211971.

G.f.: Product_{k>=1} 1/((1 - x^(10*k))*(1 - x^(10*k-1))*(1 - x^(10*k-9))). - Ilya Gutkovskiy, Aug 13 2017
a(n) ~ exp(Pi*sqrt(n/5))/(2*(sqrt(5)-1)*n). - Vaclav Kotesovec, Aug 14 2017
a(n) = a(n-1) + a(n-9) - a(n-12) - a(n-28) + + - - (with the convention a(n) = 0 for negative n), where 1, 9, 12, 28, ... is the sequence of generalized 12-gonal numbers A195162. - Peter Bala, Dec 10 2020

More terms from Omar E. Pol, Jun 10 2012

A195838 Triangle read by rows which arises from A195828, in the same way as A175003 arises from A195310. Column k starts at row A001082(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 14, -10, -1, 35, 16, -12, -2, 1, 38, 21, -13, -3, 1, 44, 32, -14, -4, 1

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. This sequence is related to the generalized octagonal numbers A001082, A195828 and A195848 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1,   1;
.  2,   1;
.  3,   1;
.  4,   1,  -1;
.  4,   1,  -1;
.  4,   2,  -1;
.  5,   3,  -1;
.  7,   4,  -1;
. 10,   4,  -2;
. 12,   4,  -3;
. 13,   5,  -4;
. 14,   7,  -4,  -1;
. 16,  10,  -4,  -1;
. 21,  12,  -5,  -1;
. 27,  13,  -7,  -1;
. 32,  14, -10,  -1;
. 35,  16, -12,  -2,   1;
. 38,  21, -13,  -3,   1;

Row sums give A195848.

Cf. A001082, A175003, A195828, A195836, A195837.

A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 1, 1, 6, 12, 19, 14, 8, 6, 1, 18, 25, 20, 6, 8, 12, 24, 43, 31, 14, 1, 8, 30, 42, 32, 1, 12, 18, 48, 73, 38, 20, 14, 6, 42, 56, 44, 12, 6, 24, 48, 91, 57, 31, 18, 14, 54, 79, 72, 8, 20, 30, 60, 114, 62, 32, 8, 1, 84, 84, 68, 18, 24, 48, 72

Offset: 1

Seiichi Manyama, Mar 25 2017

			From _Peter Bala_, Dec 11 2020: (Start)
n = 20: n is not of the form m*(3*m +- 2), so e(n) = 0 and a(20) = a(19) + a(15) - a(12) - a(4) = 20 + 6 - 19 - 1 = 6;
n = 21: n = m*(3*m - 2) for m = 3, so e(n) = 21 and a(21) = 21 + a(20) + a(16) - a(13) - a(5) = 21 + 6 + 1 - 14 - 6 = 8;
n = 40: n = m*(3*m - 2) for m = 4, so e(n) = -40 and a(4) = -40 + a(39) + a(35) - a(32) - a(24) + a(19) + a(7) = -40 + 14 + 48  - 1 - 43 + 20 + 8 = 6. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A089802 (f(-x, -x^5)), A195848 (1/f(-x, -x^5)), A222171.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), this sequence (k=6), A284363 (k=7), A284372 (k=12).

Table[Sum[If[Mod[d, 6] <2 || Mod[d, 6]==5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)

a(n) = sumdiv(n, d, ((d + 1) % 6 < 3) * d); \\ Amiram Eldar, Apr 12 2024

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1} ( (6*k)*x^(6*k)/(1 - x^(6*k)) + (6*k-1)*x^(6*k-1)/(1 - x^(6*k-1)) + (6*k-5)*x^(6*k-5)/(1 - x^(6*k-5)) ).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - -, where [1, 5, 8, 16, ...] is the sequence of generalized octagonal numbers A001082, and e(n) = (-1)^(m+1)*n if n is a generalized octagonal number of the form m*(3*m+-2); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = A222171 = 0.411233... . - Amiram Eldar, Apr 12 2024

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

Original entry on oeis.org

1, -1, -2, -2, 3, 8, 0, -2, -10, -4, 2, 4, 10, -8, -4, 0, 7, 12, 4, -2, -16, -16, 4, 8, 0, -7, -4, -2, 10, 24, 8, -2, -26, 0, 2, 8, 12, -16, -8, -8, 10, 12, 0, -6, -20, -16, 4, 8, 26, -7, -10, 0, 16, 40, 0, -4, -20, -24, 6, 4, 0, -16, -12, -8, 15, 24, 8, -6

Offset: 0

Michael Somos, Nov 26 2019

sign

			G.f. = 1 - x - 2*x^2 - 2*x^3 + 3*x^4 + 8*x^5 - 2*x^7 - 10*x^8 - 4*x^9 + ...

Cf. A028967, A030206, A195848, A224822, A329956, A329957.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^2, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^3 / eta(x^6 + A)^2, n))};

Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].
G.f.: Product_{k>=1} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k))^2.
Convolution of A030206 and A195848.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.
a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

A346797 Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 7, 4, 9, 6, 10, 11, 11, 17, 13, 22, 19, 25, 29, 28, 42, 34, 53, 46, 61, 67, 69, 92, 83, 115, 109, 133, 149, 152, 198, 182, 243, 233, 282, 309, 324, 398, 385, 485, 483, 563, 621, 648, 784, 768, 944, 947, 1096, 1194, 1262

Offset: 0

Ludovic Schwob, Aug 04 2021

nonn

			For n=17 the a(17)=6 solutions are 2+2+2+2+2+2+5, 2+2+2+2+2+7, 2+2+2+2+9, 2+5+5+5, 5+5+7 and 5+12.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000041, A047386, A274830, A006950, A036820, A036822, A195848, A035363, A195849, A346798.

CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-2))(1 - x^(7*k-5))),{k,52}],{x,0,52}],x] (* Stefano Spezia, Aug 04 2021 *)

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-2))*(1 - x^(7*k-5))).
a(n) = a(n-2) + a(n-5) - a(n-11) - a(n-17) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 2, 5, 11, 17, ... is the sequence A274830.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(3*Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 3, 6, 4, 4, 8, 9, 6, 10, 15, 12, 12, 21, 22, 18, 25, 36, 30, 32, 48, 52, 45, 60, 78, 72, 75, 105, 113, 105, 130, 166, 156, 166, 218, 236, 224, 274, 332, 325, 345, 436, 469, 462, 544, 649, 644, 688, 839, 907, 903, 1051

Offset: 0

Ludovic Schwob, Aug 04 2021

nonn

			For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.

CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* Stefano Spezia, Aug 04 2021 *)

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-3))*(1 - x^(7*k-4))).
a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A363275 Expansion of 1 / Sum_{k>=0} x^(k(3k - 2)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -7, 6, -4, 1, 3, -9, 16, -24, 32, -39, 44, -46, 44, -35, 18, 8, -43, 86, -135, 187, -238, 280, -304, 300, -259, 171, -28, -174, 435, -746, 1088, -1431, 1736, -1952, 2017, -1864, 1425, -641, -527, 2086, -4002

Offset: 0

Ilya Gutkovskiy, May 25 2023

sign

Cf. A000567, A106507, A195848, A308806, A317665, A322800, A361979, A363149.

nmax = 56; CoefficientList[Series[1/Sum[x^(k (3 k - 2)), {k, 0, nmax}], {x, 0, nmax}], x]

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)^jx^(kj*(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

Ilya Gutkovskiy, Aug 13 2017

			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, ...

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

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.

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

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)).

cp's OEIS Frontend

A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

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A195838 Triangle read by rows which arises from A195828, in the same way as A175003 arises from A195310. Column k starts at row A001082(k+1).

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A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

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A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

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A346797 Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

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A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

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A363275 Expansion of 1 / Sum_{k>=0} x^(k*(3*k - 2)).

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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).

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A363275 Expansion of 1 / Sum_{k>=0} x^(k(3k - 2)).

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)^jx^(kj*(j-1)/2 + j^2).