A261776 -id:A261776 - cp's OEIS frontend

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Ilya Gutkovskiy, May 11 2017

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Partition Function b_k
Index entries for sequences related to partitions

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
Main diagonal gives A000041.
Mirror of A061198.
Cf. A061199, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

A284344 Sum of the divisors of n that are not divisible by 10.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 8, 12, 28, 14, 24, 24, 31, 18, 39, 20, 12, 32, 36, 24, 60, 31, 42, 40, 56, 30, 32, 32, 63, 48, 54, 48, 91, 38, 60, 56, 20, 42, 96, 44, 84, 78, 72, 48, 124, 57, 33, 72, 98, 54, 120, 72, 120, 80, 90, 60, 48, 62, 96, 104, 127, 84, 144

Offset: 1

Seiichi Manyama, Mar 25 2017

nonn

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

Cf. A261776.

Cf. Sum of the divisors of n that are not divisible by k: A046913 (k=3), A046897 (k=4), A116073 (k=5), A284326 (k=6), A113957 (k=7), A284341 (k=8), A116607 (k=9), this sequence (k=10).

Table[Sum[Boole[Mod[d, 10]>0] d , {d, Divisors[n]}], {n, 100}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],Last[IntegerDigits[#]]!=0&]],{n,70}] (* Harvey P. Dale, Jun 29 2022 *)

for(n=1, 100, print1(sumdiv(n, d, ((d%10)>0)*d), ", ")) \\ Indranil Ghosh, Mar 25 2017

from sympy import divisors
print([sum([i for i in divisors(n) if i%10]) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

G.f.: Sum_{k>=1} k*x^k/(1 - x^k) - 10*k*x^(10*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Mar 25 2017

Sum_{k=1..n} a(k) ~ (3*Pi^2/40) * n^2. - Amiram Eldar, Oct 04 2022

A213598 Number of partitions of n in which no parts are multiples of 49.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524

Offset: 0

Michael Somos, Jun 14 2012

nonn

For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).

In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).

Cf. A000041, A287926.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};

Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
a(n) = (1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017

A320612 Number of parts in all partitions of n in which no part occurs more than nine times.

Original entry on oeis.org

1, 3, 6, 12, 20, 35, 54, 86, 128, 182, 264, 376, 520, 718, 978, 1318, 1761, 2338, 3070, 4008, 5206, 6707, 8604, 10982, 13933, 17604, 22155, 27745, 34627, 43061, 53338, 65859, 81074, 99458, 121687, 148469, 180633, 219202, 265386, 320473, 386147, 464245, 556925

Offset: 1

Alois P. Heinz, Oct 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Column k=9 of A210485.

Cf. A261776.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(9*i*(i+1)/2 [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 9))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);

Table[Length[  Flatten[Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]] <= 9 &]]], {n, 43}] (* Robert Price, Jul 31 2020 *)

a(n) ~ log(10) * exp(Pi*sqrt(3*n/5)) / (2 * Pi * 15^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 18 2018

A061196 If n = Sum_{i} c_i * 10^i then let b(n) = Sum_{i} c_i * (i+1). Order the integers by b(n) and then n.

Original entry on oeis.org

0, 1, 2, 10, 3, 11, 100, 4, 12, 20, 101, 1000, 5, 13, 21, 102, 110, 1001, 10000, 6, 14, 22, 30, 103, 111, 200, 1002, 1010, 10001, 100000, 7, 15, 23, 31, 104, 112, 120, 201, 1003, 1011, 1100, 10002, 10010, 100001, 1000000, 8, 16, 24, 32, 40, 105, 113, 121, 202

Offset: 0

Henry Bottomley, Apr 20 2001

This is in effect a listing of single-digit (nonnegative) solutions to b + 2c + 3d + 4e + ... = k.

The sequence can be considered as an irregular triangle listing partitions in which no part occurs more than 9 times. The row lengths are given by A261776. For example, in row 5 the value 102, corresponds to the partition 1+1+3 (= 2*1 + 0*2 + 1*3). - Andrew Howroyd, Apr 25 2023

			From _Andrew Howroyd_, Apr 25 2023: (Start)
The sequence as a triangle T(n,k) begins:
  0 | 0;
  1 | 1;
  2 | 2, 10;
  3 | 3, 11, 100;
  4 | 4, 12,  20, 101, 1000;
  5 | 5, 13,  21, 102,  110, 1001, 10000;
  6 | 6, 14,  22,  30,  103,  111,   200, 1002, 1010, 10001, 100000;
  ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..96
Rasa Smidtaite and Minvydas Ragulskis, Commentary: Multidimensional discrete chaotic maps, Front. Phys. (2022) Vol. 10, 1094240.

Cf. A000041, A028897, A261776.

With[{k = 7}, {{0}}~Join~Values@ PositionIndex[Array[Total@ MapIndexed[#1*First[#2] &, Reverse@ IntegerDigits[#]] &, 10^k]][[1 ;; k]]] // Flatten (* Michael De Vlieger, Dec 22 2022, solution only suitable for generating the data field *)

F(p)={my(v=vector(if(#p, p[#p], 1))); for(i=1, #p, v[p[i]]++); v}
row(n)={my(R=[F(p) | p<-partitions(n)]); vecsort([fromdigits(Vecrev(u)) | u<-R, vecmax(u)<=9])}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Apr 25 2023

For n < 10, a(A000070(n)) = n+1 and a(A026905(n)) = 10^(n-1).

cp's OEIS Frontend

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

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A284344 Sum of the divisors of n that are not divisible by 10.

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A213598 Number of partitions of n in which no parts are multiples of 49.

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A320612 Number of parts in all partitions of n in which no part occurs more than nine times.

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A061196 If n = Sum_{i} c_i * 10^i then let b(n) = Sum_{i} c_i * (i+1). Order the integers by b(n) and then n.

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