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 1-8 of 8 results.

A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 36, 35, 16, 5, 1, 1, 202, 201, 84, 25, 6, 1, 1, 1828, 1827, 656, 165, 36, 7, 1, 1, 27338, 27337, 8148, 1625, 286, 49, 8, 1, 1, 692004, 692003, 167568, 25509, 3396, 455, 64, 9, 1, 1, 30251722, 30251721, 5866452, 664665, 64350, 6321, 680, 81, 10, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006, corrected Dec 12 2006

Keywords

Comments

Determinant of n X n upper left submatrix is 2^[n(n-1)(n-2)/6] (see A125791). Related to partitions of numbers into powers of 2 (see A078121). Triangle A078121 shifts left one column under matrix square.

Examples

			Recurrence T(n,k) = T(n,k-1) + T(n-1,2*k) is illustrated by:
  T(4,3) = T(4,2) + T(3,6) = 201 + 455 = 656;
  T(5,3) = T(5,2) + T(4,6) = 1827 + 6321 = 8148;
  T(6,3) = T(6,2) + T(5,6) = 27337 + 140231 = 167568.
Rows of this table begin:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...;
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...;
  1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, ...;
  1, 36, 201, 656, 1625, 3396, 6321, 10816, 17361, 26500, 38841, ...;
  1, 202, 1827, 8148, 25509, 64350, 140231, 274856, 497097, ...;
  1, 1828, 27337, 167568, 664665, 2026564, 5174449, 11622976, ...;
  1, 27338, 692003, 5866452, 29559717, 109082974, 326603719, ...;
  1, 692004, 30251721, 356855440, 2290267225, 10243585092, ...; ...
Triangle A078121 begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,  10,  16,   8,   1;
  1,  36,  84,  64,  16,  1;
  1, 202, 656, 680, 256, 32, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078121 equals column 2^k-1 of this table A125790.
Matrix cube A078121^3 begins:
     1;
     3,    1;
     9,    6,    1;
    35,   36,   12,   1;
   201,  286,  144,  24,  1;
  1827, 3396, 2300, 576, 48, 1; ...
where row sums form column 3 of this table A125790,
and column 0 of A078121^3 forms column 2 of this table A125790.

Links

Crossrefs

Cf. A078121; A002577; A125791; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798; A125799 (antidiagonal sums); related table: A125800 (q=3).

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, ] = T[, 0] = 1; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 15 2015 *)
  • PARI
    {T(n,k,p=0,q=2)=local(A=Mat(1), B); if(n

Formula

T(n,k) = T(n,k-1) + T(n-1,2*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.
Conjecture: g.f. for n-th row is (Sum_{i=0..n-1} x^i Sum_{j=0..i} binomial(n+1,j)*T(n,i-j)*(-1)^j)/(1-x)^(n+1) for n > 0. - Mikhail Kurkov, May 03 2025

A125792 Column 2 of table A125790; also equals row sums of matrix power A078121^2.

Original entry on oeis.org

1, 3, 9, 35, 201, 1827, 27337, 692003, 30251721, 2320518947, 316359580361, 77477180493603, 34394869942983369, 27893897106768940835, 41603705003444309596873, 114788185359199234852802339, 588880400923055731115178072777, 5642645813427132737155703265972003
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.
Number of partitions of 2^n into powers of 2, excluding the trivial partition 2^n=2^n. - Valentin Bakoev, Feb 15 2009

Examples

			G.f.: 1 + 3*x + 9*x^2 + 35*x^3 + 201*x^4 + 1827*x^5 + 27337*x^6 + 692003*x^7 + ...
To obtain t_2(5,1) we use the table T, defined as T[i,j]= t_2(i,j), for i=1,2,...,5(=n), and j= 0,1,2,...,16(= k*m^{n-1}). It is 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 1,3,5,7,9,11,13,15,17 1,9,25,49,81 1,35,165 1,201 Column 1 contains the first 5 members of A125792. [_Valentin Bakoev_, Feb 15 2009]

Links

Crossrefs

Cf. A125790, A078121; A002575; columns: A002577, A125793, A125794, A125795, A125796; diagonals: A125797, A125798.
Adding 1 to the members of A125792 we obtain A002577. [Valentin Bakoev, Feb 15 2009]
A diagonal of A152977.

Programs

  • Maple
    g:= proc(b, n, k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1, n+1,2)-1: seq(a(n), n=0..25);  # Alois P. Heinz, Feb 27 2009
  • Mathematica
    T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, ] = T[, 0] = 1; Table[T[n, 2], {n, 0, 20} ] (* Jean-François Alcover, Jun 15 2015 *)
  • PARI
    {a(n)=local(p=2,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]))}
for(n=0,25,print1(a(n),", "))
  • PARI
    {a(n, k=3) = if(n<1, n==0, sum(i=1, k, a(n-1, 2*i-1)))}; /* Michael Somos, Nov 24 2016 */

Formula

Is this sequence the same as A002575 (coefficients of Bell's formula)?
Denote the sum m^n + m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are natural numbers). The general formula for the number of all partitions of the sum k*m^n into powers of m, smaller than m^n, is t_m(n, k)= 1 when n=1 or k=0, or = t_m(n, k-1) + Sum_{j=1..m} t_m(n-1, (k-1)*n+j), when n > 1 and k > 0. A125792 is obtained for m=2 and n=1,2,3,... [Valentin Bakoev, Feb 15 2009]
a(n) = A145515(n+1,2)-1. - Alois P. Heinz, Feb 27 2009
From Benedict W. J. Irwin, Nov 16 2016: (Start)
Conjecture: a(n+1) = Sum_{i_1=1..3} Sum_{i_2=1..2*i_1-1} ... Sum_{i_n=1..2*i_(n-1)-1} (2*i_n - 1). For example:
a(2) = Sum_{i=1..3} 2*i-1.
a(3) = Sum_{i=1..3} Sum_{j=1..2*i-1} 2*j-1.
a(4) = Sum_{i=1..3} Sum_{j=1..2*i-1} Sum_{k=1..2*j-1} 2*k-1. (End)

A125794 Column 4 of table A125790; also equals row sums of matrix power A078121^4.

Original entry on oeis.org

1, 5, 25, 165, 1625, 25509, 664665, 29559717, 2290267225, 314039061413, 77160820913241, 34317392762489765, 27859502236825957465, 41575811106337540656037, 114746581654195790543205465, 588765612737696531880325270437, 5642056933026209681424588087899225
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.

Links

Crossrefs

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125795, A125796; diagonals: A125797, A125798.
A diagonal of A152977.

Programs

  • PARI
    a(n)=local(p=4,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]))

A125799 Antidiagonal sums of table A125790.

Original entry on oeis.org

1, 2, 4, 9, 25, 94, 520, 4521, 64793, 1581010, 67106004, 5029631745, 673439168257, 162631617757086, 71416302988324776, 57430160224301687377, 85096038984339418975505, 233592305902515392375925762, 1193627868786115606927913952196, 11402285904243733254203516140245465
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; diagonals: A125797, A125798.

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^(c+1))[n-c+1,1]))
Showing 1-8 of 8 results.

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