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).

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

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

N. Robbins, On partition functions and divisor sums, J. Int. Sequences, 5 (2002) 02.1.4.

Example

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,

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 *)

Prog

(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), ", ") ; ) ; ) ; }

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.

Sequence in context: A357181 A357137 A333381 * A101950 A104562 A164306

Adjacent sequences: A124091 A124092 A124093 * A124095 A124096 A124097

Keyword

easy,nonn,tabl

Author

R. J. Mathar, Nov 26 2006

Status

approved