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 11-18 of 18 results.

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.

A124557 Main diagonal of table A124550.

Original entry on oeis.org

1, 1, 7, 91, 1899, 57876, 2447115, 139777303, 10629219251, 1066463205220, 140409644914798, 24185696469330452, 5439617764120907676, 1594552369099740836202, 608364562372792302094447
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 (table); A124558, A124559; A124551, A124552, A124554, A124555, A124556.

A124558 Secondary diagonal of table A124550; a(n) = A124550(n+1,n).

Original entry on oeis.org

1, 2, 15, 204, 4345, 133212, 5621371, 319211576, 24097683942, 2399637270890, 313606810455697, 53638534570897308, 11984755429488415041, 3491974842611221434342, 1324861497596788043284935
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 (table); A124557, A124559; A124551, A124552, A124554, A124555, A124556.

Formula

a(n) is divisible by (n+1): A124559(n) = a(n)/(n+1).

A124559 Derived from secondary diagonal of table A124550; a(n) = A124550(n+1,n)/(n+1).

Original entry on oeis.org

1, 1, 5, 51, 869, 22202, 803053, 39901447, 2677520438, 239963727089, 28509710041427, 4469877880908109, 921904263806801157, 249426774472230102453, 88324099839785869552329
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 (table); A124557, A124558; A124551, A124552, A124554, A124555, A124556.

Formula

a(n) = (n+1)*A124558(n).

A124568 Triangle, read by rows, where row n equals the inverse binomial transform of the column n in rectangular table A124550 (starting with row 1).

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 5, 25, 36, 16, 16, 143, 364, 362, 125, 66, 990, 3909, 6417, 4728, 1296, 348, 8464, 48518, 116274, 135932, 76867, 16807, 2321, 89741, 720078, 2370938, 3923330, 3441366, 1518460, 262144, 19437, 1180978, 12965026, 56627440
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2006

Keywords

Examples

			Triangle begins:
1;
1, 1;
2, 5, 3;
5, 25, 36, 16;
16, 143, 364, 362, 125;
66, 990, 3909, 6417, 4728, 1296;
348, 8464, 48518, 116274, 135932, 76867, 16807;
2321, 89741, 720078, 2370938, 3923330, 3441366, 1518460, 262144; ...

Crossrefs

Cf. A124550, A124551, A000272.

Formula

T(n,n) = (n+1)^(n-1) = A000272(n+1). T(n,0) = A124551(n).

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.

A124540 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_k(y)^n ]^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, 26, 16, 0, 1, 5, 26, 73, 107, 62, 0, 1, 6, 40, 156, 369, 486, 274, 0, 1, 7, 57, 285, 939, 1959, 2398, 1332, 0, 1, 8, 77, 470, 1995, 5764, 10912, 12668, 6978, 0, 1, 9, 100, 721, 3756, 13976, 36248, 63543, 70863, 38873, 0
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

Antidiagonal sums equal row 1 (A124531).

Examples

			Row g.f.s R_n(y) simultaneously satisfy:
R_n(y) = [1 + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...]^n
more explicitly:
R_0 = [1 + y + y^2 + y^3 + y^4 + ...]^0 = 1;
R_1 = [1 + y*(R_1)^1 + y^2*(R_2)^1 + y^3*(R_3)^1 + y^4*(R_4)^1 +...]^1;
R_2 = [1 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...]^2;
R_3 = [1 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...]^3;
R_4 = [1 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...]^4;
etc., for all rows.
Table begins:
1,0,0,0,0,0,0,0,0,0,0,...
1,1,2,5,16,62,274,1332,6978,38873,228090,...
1,2,7,26,107,486,2398,12668,70863,416304,2552490,...
1,3,15,73,369,1959,10912,63543,385341,2424988,15788469,...
1,4,26,156,939,5764,36248,233900,1549193,10529052,73390856,...
1,5,40,285,1995,13976,98665,704810,5107950,37619020,281850156,...
1,6,57,470,3756,29658,233241,1836912,14543877,116087596,936035298,...
1,7,77,721,6482,57057,495922,4282895,36922550,318834341,2765000007,...
1,8,100,1048,10474,101800,970628,9140344,85445683,795971176,7410928800,...
1,9,126,1461,16074,171090,1777416,18151272,183201255,1834958107,...
1,10,155,1970,23665,273902,3081700,33954660,368443380,3954149640,...
1,11,187,2585,33671,421179,5104528,60398327,701775756,8042277034,...
1,12,222,3316,46557,626028,8133916,102916452,1275653922,15559229828,...

Crossrefs

Rows: A124531, A124542, A124543, A124544, A124545, A124546; diagonals: A124547, A124548, A124549; related tables: A124530, A124550, A124460.

Programs

  • PARI
    T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+1)))))); Vec(Ser(R[n+1])^n+O(x^(k+1)))[k+1]

Formula

Let S_n(y) be the g.f. of row n in table A124530, then R_n(y) = S_n(y)^n and so S_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0, where R_n(y) is the g.f. of row n in this table.

A135057 Largest semiprime whose prime factors add up to 10^n.

Original entry on oeis.org

25, 2491, 249919, 24993439, 2499984871, 249999996751, 24999999992431, 2499999999940951, 249999999999995239, 24999999999999996031, 2499999999999999964279, 249999999999999999946639, 24999999999999999997538239, 2499999999999999999999854839, 249999999999999999999999946639
Offset: 1

Views

Author

Lekraj Beedassy, Feb 11 2008

Keywords

Links

Programs

  • Mathematica
    s={};f[{p_,e_}]:=e*p;Do[a=(10^n/2)^2;While[PrimeOmega[a]!=2||Total[f/@FactorInteger[a]]!=10^n,a=a-1];AppendTo[s,a],{n,11}];s (* James C. McMahon, Apr 13 2025 *)

Formula

a(n) = A124450(n)*(10^n - A124550(n)) {= A137611(n)*A137612(n) for n>1}.
a(n) = 100^n/4-(A124049(n))^2. - Zak Seidov, Feb 15 2008
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