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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A125793 Column 3 of table A125790; also equals row sums of matrix power A078121^3.

Original entry on oeis.org

1, 4, 16, 84, 656, 8148, 167568, 5866452, 356855440, 38315189204, 7352635371152, 2547660633170900, 1607532367023451792, 1860491404939092059092, 3974085151281967171382928, 15751822048486986712162264020
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125794, A125795, A125796; diagonals: A125797, A125798.

Programs

  • PARI
    a(n)=local(p=3,q=2,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125795 Column 5 of table A125790; also equals row sums of matrix power A078121^5.

Original entry on oeis.org

1, 6, 36, 286, 3396, 64350, 2026564, 109082974, 10243585092, 1704787839326, 509106367263812, 275575947307878750, 272638898948894782532, 496470192421055920965982, 1674003944602430578138969156, 10505662319550964196499807897950, 123269344114733507237294056110191684
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125796; diagonals: A125797, A125798.

Programs

  • PARI
    a(n)=local(p=5,q=2,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125796 Column 6 of table A125790; also equals row sums of matrix power A078121^6.

Original entry on oeis.org

1, 7, 49, 455, 6321, 140231, 5174449, 326603719, 35994670257, 7036275790791, 2470183452677297, 1573137497080468423, 1832597507832323118257, 3932481446278522861786055, 15637033863127787477309461681, 115814953429924513361085880079303, 1604893891765170672173387008222303409
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795; diagonals: A125797, A125798.

Programs

  • PARI
    a(n)=local(p=6,q=2,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125797 Main diagonal of table A125790.

Original entry on oeis.org

1, 2, 9, 84, 1625, 64350, 5174449, 841185704, 275723872209, 181906966455026, 241258554545388985, 642662865556736504700, 3436011253857466940820073, 36852501476559726217536067974, 792571351187806816558255494473185
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125798 (diagonal).

Programs

  • PARI
    a(n)=local(q=2,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^n)[n+1,c+1]))

A125798 A diagonal of table A125790: a(n) = A125790(n+1,n).

Original entry on oeis.org

1, 4, 35, 656, 25509, 2026564, 326603719, 106355219008, 69808185542089, 92203545302072964, 244779396712068825067, 1305009502037405316440848, 13963029918525356899170492525, 299675759834305402824238609624548
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125797 (diagonal).

Programs

  • PARI
    a(n)=local(q=2,A=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n+1,(A^n)[n+2,c+1]))

A111826 Number of partitions of 5*6^n into powers of 6, also equals column 1 of triangle A111825, which shifts columns left and up under matrix 6th power.

Original entry on oeis.org

1, 6, 96, 6306, 1883076, 2700393702, 19324893252552, 709398600017820522, 136347641698786289641932, 139389318443495655514432423662, 767442745549858935398537400096197328
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=6; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Links

Crossrefs

Cf. A111825 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111831 (q=7), A111836 (q=8).

Programs

  • PARI
    a(n,q=6)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

Formula

a(n) = [x^(5*6^n)] 1/Product_{j>=0}(1-x^(6^j)).

A111831 Number of partitions of 6*7^n into powers of 7, also equals column 1 of triangle A111830, which shifts columns left and up under matrix 7th power.

Original entry on oeis.org

1, 7, 154, 16275, 9106461, 28543862991, 521136519414483, 56980036448207052005, 38084892600214854893482284, 158081960770204032330986210466109, 4125860571927530263431055188002578191656
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=7; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Links

Crossrefs

Cf. A111830 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111826 (q=6), A111836 (q=8).

Programs

  • PARI
    a(n,q=7)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

Formula

a(n) = [x^(6*7^n)] 1/Product_{j>=0}(1-x^(7^j)).

A111836 Number of partitions of 7*8^n into powers of 8, also equals column 1 of triangle A111835, which shifts columns left and up under matrix 8th power.

Original entry on oeis.org

1, 8, 232, 36968, 35593832, 219379963496, 9003699178010216, 2530260913162860295784, 4970141819535151534947497576, 69322146154435681317709098939119208
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=8; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Links

Crossrefs

Cf. A111835 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111821 (q=5), A111826 (q=6), A111831 (q=7).

Programs

  • PARI
    a(n,q=8)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

Formula

a(n) = [x^(7*8^n)] 1/Product_{j>=0}(1-x^(8^j)).

A111817 Number of partitions of 3*4^n into powers of 4, also equals column 1 of triangle A078536, which shifts columns left and up under matrix 4th power.

Original entry on oeis.org

1, 4, 28, 524, 29804, 5423660, 3276048300, 6744720496300, 48290009081437356, 1221415413140406958252, 110523986015743458745929900, 36150734459755630877180158951596
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Comments

Let q=4; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Links

Crossrefs

Cf. A078536 (triangle), A002577 (q=2), A078124 (q=3), A111821 (q=5), A111826 (q=6), A111831 (q=7), A111836 (q=8).

Programs

  • PARI
    a(n,q=4)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+2,2]))

Formula

a(n) = [x^(3*4^n)] 1/Product_{j>=0}(1-x^(4^j)).

A111822 Number of partitions of 5^n into powers of 5, also equals the row sums of triangle A111820, which shifts columns left and up under matrix 5th power.

Original entry on oeis.org

1, 2, 7, 82, 3707, 642457, 446020582, 1288155051832, 15905066118254957, 856874264098480364332, 204616369654716156089739332, 219286214391142987407272329973707, 1065403165201779499307991460987124895582, 23663347632778954225192551079067428619449114332
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Links

Crossrefs

Cf. A111820, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111827 (q=6), A111832 (q=7), A111837 (q=8).
Column k=5 of A145515.

Programs

  • PARI
    a(n,q=5)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))

Formula

a(n) = [x^(5^n)] 1/Product_{j>=0}(1-x^(5^j)).
Previous Showing 21-30 of 41 results. Next

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