A036820 -id:A036820 - 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

A195837 Triangle read by rows which arises from A195827, in the same way as A175003 arises from A195310. Column k starts at row A085787(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 5, -3, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 14, -7, -1, 33, 16, -10, -2, 37, 21, -12, -3, 1, 44, 27, -14, -4, 1, 54, 33, -16, -4, 1, 68, 37, -21, -5, 1, 80, 44, -27, -7, 2

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 heptagonal numbers A085787, A195827 and A036820 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. It appears that row sums give A036820. See comments in A195825.

			Written as a triangle:
.  1;
.  1;
.  1;
.  1,   1;
.  2,   1;
.  3,   1;
.  4,   1,  -1;
.  4,   2,  -1;
.  5,   3,  -1;
.  7,   4,  -1;
. 10,   4,  -2;
. 12,   5,  -3;
. 14,   7,  -4,  -1;
. 16,  10,  -4,  -1;
. 21,  12,  -5,  -1;
. 27,  14,  -7,  -1;
. 33,  16, -10,  -2;
. 37,  21, -12,  -3,  1;
. 44,  27, -14,  -4,  1;
. 54,  33, -16,  -4,  1;

Cf. A085787, A036820, A175003, A195825, A195826, A195827, A195836, A195838, A195839, A195840, A195841, A195842, A195843.

A195827 Triangle read by rows with T(n,k) = n - A085787(k), n>=1, k>=1, if (n - A085787(k))>=0.

Original entry on oeis.org

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

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 A085787(k).

This sequence is related to the generalized heptagonal numbers A085787, A195837 and A036820 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,  0;
.  4,  1;
.  5,  2;
.  6,  3,  0;
.  7,  4,  1;
.  8,  5,  2;
.  9,  6,  3;
. 10,  7,  4;
. 11,  8,  5;
. 12,  9,  6,  0;
. 13, 10,  7,  1;
. 14, 11,  8,  2;

Cf. A036820, A085787, A195310, A195825, A195826, A195828, A195829, A195830, A195831, A195832, A195833, A195837.

A301567 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 4, 1, 0, 2, 7, 7, 2, 0, 3, 10, 11, 4, 0, 4, 14, 17, 8, 1, 5, 19, 25, 13, 2, 6, 25, 36, 21, 4, 8, 33, 50, 33, 8, 10, 43, 69, 49, 14, 13, 55, 93, 71, 23, 17, 70, 124, 102, 37, 22, 88, 163, 142, 57, 30, 110, 212, 195, 85

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 5.

			a(11) = 3 because we have [11], [10, 1] and [6, 5].

Index entries for sequences related to partitions

Cf. A008851, A035367, A036820, A203776, A219607, A280454, A301562, A301563, A301564, A301565, A301568, A301569, A301570.

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[x^4 QPochhammer[-1, x^5] QPochhammer[-x^(-4), x^5]/(2 (1 + x^4)), {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A008851(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301570 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 2, 0, 0, 4, 7, 3, 0, 1, 7, 10, 4, 0, 2, 11, 14, 5, 0, 4, 17, 19, 6, 0, 8, 25, 25, 8, 1, 13, 36, 33, 10, 2, 21, 50, 43, 12, 4, 33, 69, 55, 15, 8, 49, 93, 70, 18, 14, 71, 124, 88, 23, 23, 102, 163, 110, 29, 37

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 4 mod 5.

			a(14) = 3 because we have [14], [10, 4] and [9, 5].

Index entries for sequences related to partitions

Cf. A035370, A036820, A047208, A203776, A219607, A281243, A301562, A301563, A301564, A301565, A301567, A301568, A301569.

nmax = 76; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[x QPochhammer[-1, x^5] QPochhammer[-x^(-1), x^5]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A047208(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(41/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 1, 5, 10, 16, 12, 11, 1, 15, 21, 21, 1, 16, 20, 40, 22, 12, 1, 35, 31, 27, 10, 19, 30, 67, 32, 21, 12, 35, 41, 56, 1, 20, 40, 80, 42, 42, 1, 60, 75, 47, 1, 51, 50, 91, 52, 31, 1, 70, 72, 75, 20, 30, 60, 151, 62, 32, 31, 85, 71, 84, 1, 39, 70, 135

Offset: 1

Seiichi Manyama, Mar 25 2017

nonn

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

Cf. A036820 (1/f(-x, -x^4)), A113429 (f(-x, -x^4)), A102753.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), this sequence (k=5), A284362 (k=6), A284363 (k=7), A284372 (k=12).

Table[Sum[If[Mod[d, 5]<2 || Mod[d, 5]==4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],MemberQ[{0,1,4},Mod[#,5]]&]],{n,70}] (* Harvey P. Dale, Aug 02 2020 *)

a(n) = sumdiv(n, d, ((d + 1) % 5 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = A102753 / 10 = 0.4934802... . - Amiram Eldar, Apr 12 2024

A036822 Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 4, 7, 6, 10, 11, 13, 18, 19, 25, 30, 33, 45, 47, 61, 70, 81, 100, 111, 135, 157, 177, 218, 238, 288, 328, 374, 443, 495, 579, 663, 747, 878, 973, 1134, 1281, 1448, 1670, 1863, 2135, 2414, 2705, 3103

Offset: 1

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (1=4 := 0).
a(n) is the number of partitions with parts congruent to 0, 2 or 3 mod 5. - George Beck, Aug 08 2020

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A036820.

c := proc(L,i,n)
    local a,p;
    a := 0 ;
    for p in L do
        if modp(p,n) = i then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
A036822 := proc(n)
    local a ,p;
    a := 0 ;
    for p in combinat[partition](n) do
        if c(p,1,5) = 0 then
            if c(p,4,5) = 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 19 2014

nmax = 50; Rest[CoefficientList[Series[Product[1/((1 - x^(5*k)) * (1 - x^(5*k-2)) * (1 - x^(5*k-3))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2016 *)

Convolution inverse of A113428. - George Beck, May 21 2016
G.f.: Product_{k>=1} 1/((1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3))). - Vaclav Kotesovec, Jul 05 2016
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2*sqrt(2*(5+sqrt(5)))*n). - Vaclav Kotesovec, Jul 05 2016

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

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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A195837 Triangle read by rows which arises from A195827, in the same way as A175003 arises from A195310. Column k starts at row A085787(k).

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A195827 Triangle read by rows with T(n,k) = n - A085787(k), n>=1, k>=1, if (n - A085787(k))>=0.

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A301567 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-4)).

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A301570 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-1)).

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A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

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PARI

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A036822 Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

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Maple

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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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A301567 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-4)).

A301570 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-1)).

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).