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.

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A124566 Row 6 of table A124560; also, the self-convolution 6th power equals A124556, which is row 6 of table A124550.

Original entry on oeis.org

1, 1, 7, 70, 891, 14036, 272914, 6525900, 190604859, 6781448755, 294798563020, 15737487680990, 1036588563202854, 84606134756948277, 8587502188940359207, 1086820294948914428468, 171866738763640156327659
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Crossrefs

Cf. A124560 (table); other rows: A124551, A124562, A124563, A124564, A124565.

Formula

G.f.: A(x) = Sum_{k>=0} y^k * [R_{6k}(y)]^(6k), where R_n(x) is the g.f. of row n in table A124560.

A124561 Antidiagonal sums of rectangular table A124560.

Original entry on oeis.org

1, 2, 3, 5, 11, 35, 159, 979, 7870, 80898, 1050628, 17117967, 348650347, 8866137878, 281540044623, 11172327581631, 554607985588213, 34475080278856486, 2685981999907232301, 262507094271746005965, 32206094635505510590194
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Crossrefs

Cf. A124560.

A124567 Main diagonal of table A124560.

Original entry on oeis.org

1, 1, 3, 22, 317, 7581, 272914, 13975298, 981599065, 92344174075, 11474433841581, 1869937363283342, 398202456366104555, 110561615797436467938, 39978343388337757968279, 18814183279605303823720601
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Crossrefs

Cf. A124560; A124557 (variant).

A124550 Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0, with R_0(y)=1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 16, 0, 1, 5, 26, 91, 159, 66, 0, 1, 6, 40, 204, 666, 1056, 348, 0, 1, 7, 57, 385, 1899, 5955, 8812, 2321, 0, 1, 8, 77, 650, 4345, 21180, 65794, 92062, 19437, 0, 1, 9, 100, 1015, 8616, 57876, 287568, 901881
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

Antidiagonal sums equal row 1 (A124551).

Examples

			The g.f. of row n, R_n(y), simultaneously satisfies:
R_n(y) = [1 + y*R_{n}(y) + y^2*R_{2n}(y) + y^3*R_{3n}(y) +...]^n
more explicitly,
R_0 = [1 + y + y^2 + y^3 +... ]^0 = 1,
R_1 = [1 + y*R_1 + y^2*R_2 + y^3*R_3 + y^4*R_4 +...]^1,
R_2 = [1 + y*R_2 + y^2*R_4 + y^3*R_6 + y^4*R_8 +...]^2,
R_3 = [1 + y*R_3 + y^2*R_6 + y^3*R_9 + y^4*R_12 +...]^3,
R_4 = [1 + y*R_4 + y^2*R_8 + y^3*R_12 + y^4*R_16 +...]^4,
etc., for all rows.
Table begins:
1,0,0,0,0,0,0,0,0,0,...
1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
1,2,7,30,159,1056,8812,92062,1200415,19512990,395379699,9991017068,...
1,3,15,91,666,5955,65794,901881,15346419,324465907,8535776700,...
1,4,26,204,1899,21180,287568,4802716,99084889,2531896840,...
1,5,40,385,4345,57876,926340,18088835,434349525,12879458545,...
1,6,57,650,8616,133212,2447115,54419202,1481595429,49675372516,...
1,7,77,1015,15449,271677,5621371,139777303,4236941723,157754261392,...
1,8,100,1496,25706,506376,11637540,319211576,10629219251,...
1,9,126,2109,40374,880326,22228296,665618589,24097683942,...
1,10,155,2870,60565,1447752,39814650,1290831110,50395939380,...
1,11,187,3795,87516,2275383,67666852,2359273213,98672395096,...
1,12,222,4900,122589,3443748,110082100,4104444564,182882370066,...
1,13,260,6201,167271,5048472,172579056,6848496031,323591733868,...
1,14,301,7714,223174,7201572,262109169,11025158762,550236760920,...
1,15,345,9455,292035,10032753,387284805,17206288875,903909656190,...
1,16,392,11440,375716,13690704,558624184,26132289904,1440743993738,...
1,17,442,13685,476204,18344394,788813124,38746675145,2235979092419,...
1,18,495,16206,595611,24184368,1092983592,56235032046,3388787136045,...
1,19,551,19019,736174,31424043,1489009062,80068650785,5027951628273,...
1,20,610,22140,900255,40301004,1997816680,112053079180,7318490555455,...
1,21,672,25585,1090341,51078300,2643716236,154381866075,10469322413655,..
1,22,737,29370,1309044,64045740,3454745943,209695755346,14742078039007,..
1,23,805,33511,1559101,79521189,4463035023,281147592671,20461165963557,..
1,24,876,38024,1843374,97851864,5705183100,372473207208,28025203801701,..

Crossrefs

Rows: A124551, A124552, A124553, A124554, A124555, A124556; diagonals: A124557, A124558, A124559; variants: A124560, A124460, A124530, A124540.

Programs

  • PARI
    {T(n,k)=if(k==0,1,if(n==0,0,if(k==1,n,if(n<=k, Vec(( 1+x*Ser( vector(k,j,sum(i=0,j-1,T(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,T(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))}

Formula

Let G_n(y) be the g.f. of row n in table A124560, then R_n(y) = G_n(y)^n and thus G_n(y) = Sum_{k>=0} y^k * R_{n*k}(y) for n>=0, where R_n(y) is the g.f. of row n in this table.
Previous Showing 11-14 of 14 results.

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