0,6

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

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

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,

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

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

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

easy,nonn,tabl

R. J. Mathar, Nov 26 2006

approved