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.

A332311 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 5.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 11, 19, 44, 31, 61, 87, 117, 144, 279, 311, 389, 541, 640, 1003, 1225, 2145, 2493, 3452, 3507, 5417, 6671, 8821, 11580, 17959, 21043, 26289, 34797, 41536, 59637, 72707, 85871, 110947, 172472, 175873, 249691, 327801, 418779, 512748
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 09 2020

Keywords

Examples

			a(6) = 9 because we have [6], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Links

Crossrefs

Cf. A032020, A032021, A035959, A096938, A332309, A332310.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 5], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 819, 3389, 14131, 59341, 250703, 1064207, 4535091, 19390229, 83139955, 357354213, 1539272499, 6642769925, 28714955571, 124312591469, 538895612751, 2338948779320, 10162837993377, 44202371860240, 192431323820851, 838442649862701, 3656031108325651
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Links

Crossrefs

Cf. A096938, A255526, A285291, A296163, A296164, A304628.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(5 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[1/(1 - x^k + x^(2 k) - x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
(* Calculation of constants {d,c}: *) With[{k = 5}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s] / QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k + x^(2*k) - x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.445766346387064439086120427... and c = 0.267035948020079842478289... - Vaclav Kotesovec, May 18 2018

A124094 Table T(n,m) giving number of partitions of n such that all parts are coprime to m. Read along antidiagonals (increasing n, decreasing m).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 5, 1, 1, 1, 2, 2, 7, 1, 1, 2, 2, 4, 3, 11, 1, 1, 1, 3, 2, 5, 4, 15, 1, 1, 2, 1, 5, 3, 7, 5, 22, 1, 1, 1, 3, 1, 6, 4, 9, 6, 30, 1, 1, 2, 2, 5, 2, 10, 5, 13, 8, 42, 1, 1, 1, 2, 2, 7, 2, 13, 6, 16, 10, 56, 1, 1, 2, 2, 4, 3, 11, 3, 19, 8, 22, 12, 77, 1, 1, 1, 3, 2, 5, 4
Offset: 0

Views

Author

R. J. Mathar, Nov 26 2006

Keywords

Examples

			Upper left corner of table starts with row m=1 and column n=0:
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243, 297, 355,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,3,5,6,10,13,19,25,34,44,60, 76,100,127,164,205,262,325,409,505, 628, 769,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,14,21,28,39,51,70, 90,119,153,199,252,324,406,515,642, 804, 994,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243, 297, 355,
1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41,  47,  54,
1,1,2,3,5,7,11,15,22,30,42,55,76, 99,132,171,224,286,370,468,597,750, 945,1177,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,15,22,30,42,56,77,100,134,174,228,292,378,479,612,770, 972,1213,
1,1,1,2,2,3, 4, 4, 5, 7, 8,10,12, 14, 17, 21, 24, 28, 34, 39, 46, 53,  61,  71,
1,1,2,2,4,4, 6, 7,11,12,16,19,25, 29, 37, 44, 56, 65, 80, 94,114,133, 160, 187,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,296,384,488,624,787, 995,1244,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,489,626,790, 999,1250,
1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41,  47,  54,

Links

Crossrefs

Row m=1 is A000041. Rows m=2, 4, 8, ... (where m is a power of 2) are A000009. Rows m=3, 9, ... (where m is a power of 3) are A000726. Row m=5 is A035959. Row=7 is A035985. Row m=10 is A096938.

Programs

  • Maple
    b:= proc(n, i, m) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-1, m) +`if`(igcd(m, i)=1, b(n-i, i, m), 0)
      fi
    end:
T:= (n, m)-> b(n, n, m):
seq (seq (T(n, 1+d-n), n=0..d), d=0..13);  # Alois P. Heinz, Sep 28 2011
  • Mathematica
    b[n_, i_, m_] := b[n, i, m] = Which[n < 0, 0, n == 0, 1, i < 1, 0, True, b[n, i-1, m] + If[GCD[m, i] == 1, b[n-i, i, m], 0]]; t[n_, m_] := b[n, n, m]; Table[Table[t[n, 1+d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Alois P. Heinz's Maple code *)
  • PARI
    sigmastar(n,m)= { local(d,res=0) ; d=divisors(n) ; for(i=1,matsize(d)[2], if( gcd(d[i],m)==1, res += d[i] ; ) ; ) ; return(res) ; } f(n,m)= { local(qvec=vector(n+1,i,gcd(1,m))) ; for(i=1,n, qvec[i+1]=sum(k=1,i,sigmastar(k,m)*qvec[i-k+1])/i ; ) ; return(qvec[n+1]) ; } { for(d=1,18, for(c=0,d-1, r=d-c ; print1(f(c,r),",") ; ) ; ) ; }
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.