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.

Showing 1-10 of 16 results. Next

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.

A124560 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*k) for n>=0, with R_0(y)=1/(1-y).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 16, 1, 1, 1, 5, 22, 63, 66, 1, 1, 1, 6, 35, 158, 429, 348, 1, 1, 1, 7, 51, 317, 1455, 3716, 2321, 1, 1, 1, 8, 70, 556, 3634, 16918, 40272, 19437, 1, 1, 1, 9, 92, 891, 7581, 52199, 244644, 541655, 203554, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Examples

			The g.f. of row n, R_n(y), simultaneously satisfies:
R_n(y) = 1 + y*R_{n}(y)^n + y^2*R_{2n}(y)^(2n) + y^3*R_{3n}(y)^(3n) +...
more explicitly,
R_0 = 1 + y + y^2 + y^3 +... = 1/(1-y),
R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 +...,
R_2 = 1 + y*(R_2)^2 + y^2*(R_4)^4 + y^3*(R_6)^6 + y^4*(R_8)^8 +...,
R_3 = 1 + y*(R_3)^3 + y^2*(R_6)^6 + y^3*(R_9)^9 + y^4*(R_12)^12 +...,
R_4 = 1 + y*(R_4)^4 + y^2*(R_8)^8 + y^3*(R_12)^12 + y^4*(R_16)^16 +...,
etc., for all rows.
Table begins:
1,1,1,1,1,1,1,1,1,1,...
1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
1,1,3,12,63,429,3716,40272,541655,9022405,186233087,4771577072,...
1,1,4,22,158,1455,16918,244644,4361883,95746603,2592416878,...
1,1,5,35,317,3634,52199,928608,20282765,543008771,17866390922,...
1,1,6,51,556,7581,128532,2689248,68880819,2155007000,82603481941,...
1,1,7,70,891,14036,272914,6525900,190604859,6781448755,...
1,1,8,92,1338,23864,521662,13975298,456468525,18121964864,...
1,1,9,117,1913,38055,921709,27263527,981599065,42880525630,...
1,1,10,145,2632,57724,1531900,49474783,1941904513,92344174075,...
1,1,11,176,3511,84111,2424288,84736940,3594121407,184465174294,...
1,1,12,210,4566,118581,3685430,138423924,6299505191,346530455866,...
1,1,13,247,5813,162624,5417683,217374894,10551425445,618507018238,...
1,1,14,287,7268,217855,7740500,330130230,17007128087,1057156741967,...
1,1,15,330,8947,286014,10791726,487184328,26523926691,1741018836674,...
1,1,16,376,10866,368966,14728894,701255202,40200085065,2776362938533,..
1,1,17,425,13041,468701,19730521,987570893,59420653233,4304220653087,..

Crossrefs

Rows: A124551, A124562, A124563, A124564, A124565, A124566; diagonals: A124567, A124568, A124569; A124561 (antidiagonal sums); variants: A124550, A124460, A124530, A124540.

Programs

  • PARI
    {A124550(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,A124550(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,A124550(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))} /* Determined Elements from A124550: */ {T(n,k)=if(n==0|k==0,1,Vec((Ser(vector(k+1,j,A124550(n,j-1)))+x*O(x^k))^(1/n))[k+1])}

Formula

Let F_n(y) be the g.f. of row n in table A124550, then F_n(y) = R_n(y)^n and thus R_n(y) = Sum_{k>=0} y^k * F_{n*k}(y) for n>=0, where R_n(y) is the g.f. of row n in this table.

A124552 Row 2 of table A124550; also equals the self-convolution square of A124562, which is row 2 of table A124560.

Original entry on oeis.org

1, 2, 7, 30, 159, 1056, 8812, 92062, 1200415, 19512990, 395379699, 9991017068, 315094522052, 12413464676162, 611490149713956, 37699912801819870, 2911578809929672737, 281916836769424155940, 34249052273023310929439
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Crossrefs

Cf. A124550; other rows: A124551, A124553, A124554, A124555, A124556.

Formula

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{2k}(y) ]^2, where R_n(x) is the g.f. of row n in table A124550.

A124554 Row 4 of table A124550; also equals the self-convolution 4th power of A124564, which is row 4 of table A124560.

Original entry on oeis.org

1, 4, 26, 204, 1899, 21180, 287568, 4802716, 99084889, 2531896840, 80346294380, 3173108251044, 156222183969181, 9603287701405136, 738066706464107408, 71003673625149131020, 8559188942710590217013, 1294012524894298022238700
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Crossrefs

Cf. A124550; other rows: A124551, A124552, A124553, A124555, A124556.

Formula

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{4k}(y) ]^4, where R_n(x) is the g.f. of row n in table A124550.

A124555 Row 5 of table A124550; also equals the self-convolution 5th power of A124565, which is row 5 of table A124560.

Original entry on oeis.org

1, 5, 40, 385, 4345, 57876, 926340, 18088835, 434349525, 12879458545, 473368667181, 21628231535280, 1231043822730950, 87448787189791250, 7764880712865963895, 862911010853538242176, 120154448960730327164325
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Crossrefs

Cf. A124550; other rows: A124551, A124552, A124553, A124554, A124556.

Formula

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{5k}(y) ]^5, where R_n(x) is the g.f. of row n in table A124550.

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

Original entry on oeis.org

1, 6, 57, 650, 8616, 133212, 2447115, 54419202, 1481595429, 49675372516, 2060520991653, 106132616602416, 6805336245809868, 544338689393810406, 54409320120862446624, 6805431454912335217590, 1066433786673596566035777
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Crossrefs

Cf. A124550; other rows: A124551, A124552, A124553, A124554, A124555.

Formula

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{6k}(y) ]^6, where R_n(x) is the g.f. of row n in table A124550.

A124553 Row 3 of table A124550; also equals the self-convolution cube of A124563, which is row 3 of table A124560.

Original entry on oeis.org

1, 3, 15, 91, 666, 5955, 65794, 901881, 15346419, 324465907, 8535776700, 279761994750, 11438401220798, 584130591952902, 37300812251856828, 2981666324471342976, 298639111371493692105, 37510558656296021069859
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Crossrefs

Cf. A124550; other rows: A124551, A124552, A124554, A124555, A124556.

Formula

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{3k}(y) ]^3, where R_n(x) is the g.f. of row n in table A124550.

A124562 Row 2 of table A124560; also, the self-convolution square equals A124552, which is row 2 of table A124550.

Original entry on oeis.org

1, 1, 3, 12, 63, 429, 3716, 40272, 541655, 9022405, 186233087, 4771577072, 152050410552, 6037181967007, 299176055730253, 18530408350215038, 1436276193993882859, 139462485162718679221, 16980510121067921518710
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, A124563, A124564, A124565, A124566.

Formula

G.f.: A(x) = Sum_{k>=0} y^k * [R_{2k}(y)]^(2k), where R_n(x) is the g.f. of row n in table A124560.

A124563 Row 3 of table A124560; also, the self-convolution cube equals A124553, which is row 3 of table A124550.

Original entry on oeis.org

1, 1, 4, 22, 158, 1455, 16918, 244644, 4361883, 95746603, 2592416878, 86825876398, 3607980811184, 186517056299848, 12022039151786966, 967930783674722085, 97490785254375447744, 12299113367136495834881, 1945525636812103772634604
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, A124564, A124565, A124566.

Formula

G.f.: A(x) = Sum_{k>=0} y^k * [R_{3k}(y)]^(3k), where R_n(x) is the g.f. of row n in table A124560.

A124564 Row 4 of table A124560; also, the self-convolution 4th power equals A124554, which is row 4 of table A124550.

Original entry on oeis.org

1, 1, 5, 35, 317, 3634, 52199, 928608, 20282765, 543008771, 17866390922, 725141498506, 36439677332431, 2274854774699772, 176901777097180896, 17172973623970540220, 2084722855480792814909, 316912193519759976734613
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, A124565, A124566.

Formula

G.f.: A(x) = Sum_{k>=0} y^k * [R_{4k}(y)]^(4k), where R_n(x) is the g.f. of row n in table A124560.
Showing 1-10 of 16 results. Next

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