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 27 results. Next

A244119 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).

Original entry on oeis.org

1, 0, 1, 0, -2, 3, 0, 3, -18, 16, 0, -4, 72, -192, 125, 0, 5, -240, 1440, -2500, 1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, 7, -2016, 45360, -280000, 680400, -705894, 262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
Sequence A161628, arising from a different context, appears to be the same, but with opposite signs of odd rows.

Examples

			First rows of the triangle, all summing up to 1:
1
0  1
0 -2    3
0  3  -18   16
0 -4   72 -192   125
0  5 -240 1440 -2500 1296

Links

Crossrefs

Cf. A161628, A244116, A244117, A244118, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
Cf. A273954.

Programs

  • Maple
    A244119 := (n, k) -> (1+k)^(k-1)*(-k)^(n-k)*binomial(n,k):
seq(seq(A244119(n, k), k = 0..n), n = 0..8); # Peter Luschny, Jan 29 2023
  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k]=(1-k*b)^(k-1)*(k*b)^(n-k)*binomial(n,k););
  );return(v);}
  a=seq(100,-1);

A244116 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 0, 1, -2, 4, 0, 1, -4, 12, -27, 0, 1, -8, 36, -108, 256, 0, 1, -16, 108, -432, 1280, -3125, 0, 1, -32, 324, -1728, 6400, -18750, 46656, 0, 1, -64, 972, -6912, 32000, -112500, 326592, -823543, 0, 1, -128, 2916, -27648, 160000, -675000, 2286144, -6588344, 16777216
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k) = (1-k)^(k-1) * k^(n-k) for k>0, and T(n,0) = 0^n by convention.

Examples

			The first few rows of the triangle are:
  1
  0 1
  0 1 -1
  0 1 -2 4
  0 1 -4 12  -27
  0 1 -8 36 -108 256
  ...

Links

Crossrefs

Cf. A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
Cf. A357247.

Programs

  • Maple
    A244116 := (n, j) -> (-1)^(j + 1) * j^(n - j) * (j - 1)^(j - 1):
for n from 0 to 9 do seq(A244116(n, k), k = 0..n) od; # Peter Luschny, Jan 28 2023
  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k););
  );return(v);}
  a=seq(100,1);

A244117 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).

Original entry on oeis.org

1, 0, 1, 0, 2, -1, 0, 3, -6, 4, 0, 4, -24, 48, -27, 0, 5, -80, 360, -540, 256, 0, 6, -240, 2160, -6480, 7680, -3125, 0, 7, -672, 11340, -60480, 134400, -131250, 46656, 0, 8, -1792, 54432, -483840, 1792000, -3150000, 2612736, -823543, 0, 9, -4608, 244944, -3483648, 20160000, -56700000, 82301184, -59295096, 16777216
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1-k)^(k-1)*k^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.

Examples

			First rows of the triangle, all summing up to 1:
1
0 1
0 2  -1
0 3  -6   4
0 4 -24  48  -27
0 5 -80 360 -540 256

Links

Crossrefs

Cf. A244116, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k)*binomial(n,k););
  );return(v);}
  a=seq(100,1);

A244118 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

Original entry on oeis.org

1, 0, 1, 0, -1, 3, 0, 1, -6, 16, 0, -1, 12, -48, 125, 0, 1, -24, 144, -500, 1296, 0, -1, 48, -432, 2000, -6480, 16807, 0, 1, -96, 1296, -8000, 32400, -100842, 262144, 0, -1, 192, -3888, 32000, -162000, 605052, -1835008, 4782969, 0, 1, -384, 11664, -128000, 810000, -3630312, 12845056, -38263752, 100000000
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k) = (1+k)^(k-1)*(-k)^(n-k) for k>0, where T(n,0) = 0^n.

Examples

			The first rows of the triangle are:
1
0  1
0 -1   3
0  1  -6  16
0 -1  12 -48  125
0  1 -24 144 -500 1296

Links

Crossrefs

Cf. A244116, A244117, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k););
  );return(v);}
  a=seq(100,-1);

A244121 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

Original entry on oeis.org

1, 0, 1, 0, 4, 0, 0, 9, 18, 0, 0, 16, 192, 48, 0, 0, 25, 1200, 1800, 100, 0, 0, 36, 5760, 29160, 11520, 180, 0, 0, 49, 23520, 317520, 423360, 58800, 294, 0, 0, 64, 86016, 2721600, 9175040, 4536000, 258048, 448, 0, 0, 81, 290304, 19840464, 145152000, 181440000, 39680928, 1016064, 648, 0
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=n*(n-k)^(k-1)*k^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.

Examples

			First rows of the triangle, all summing up to n^n:
1
0 1
0 4   0
0 9  18   0
0 16 192  48  0
0 25 1200 1800 100 0

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
  for(k=1,n,v[irow+k]=n*(n-k*b)^(k-1)*(k*b)^(n-k)*binomial(n,k);););
  return(v);}
  a=seq(100,1);

A244122 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k)*binomial(n,k).

Original entry on oeis.org

1, 0, 1, 0, -2, 8, 0, 3, -30, 108, 0, -4, 96, -588, 2048, 0, 5, -280, 2880, -14580, 50000, 0, -6, 768, -13122, 96000, -439230, 1492992, 0, 7, -2016, 56700, -596288, 3628800, -15594306, 52706752, 0, -8, 5120, -235224, 3538944, -28561000, 154893312, -637875000, 2147483648, 0
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=n*(n+k)^(k-1)*(-k)^(n-k) for k>0, while T(n,0)=0^n by convention.

Examples

			The first rows of the triangle are:
1
0  1
0 -2    8
0  3  -30  108
0 -4   96 -588   2048
0  5 -280 2880 -14580 50000

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax, b)={my(v, n, k, irow);
  v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
  for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k] = n*(n-k*b)^(k-1)*(k*b)^(n-k); ); );
  return(v); }
  a=seq(100,-1);

A244123 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

Original entry on oeis.org

1, 0, 1, 0, -4, 8, 0, 9, -90, 108, 0, -16, 576, -2352, 2048, 0, 25, -2800, 28800, -72900, 50000, 0, -36, 11520, -262440, 1440000, -2635380, 1492992, 0, 49, -42336, 1984500, -20870080, 76204800, -109160142, 52706752, 0, -64, 143360, -13172544, 247726080, -1599416000, 4337012736, -5103000000, 2147483648
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=n*(n+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.

Examples

			First rows of the triangle, all summing up to n^n:
1
0 1
0 -4  8
0, 9 -90 108
0 -16 576 -2352 2048
0, 25 -2800 28800 -72900 50000

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax, b)={my(v, n, k, irow);
  v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
  for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k]=n*(n-k*b)^(k-1)*(k*b)^(n-k)*binomial(n, k); ); );
  return(v); }
  a=seq(100,-1);

A244124 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).

Original entry on oeis.org

0, 0, 1, 0, 2, -1, 0, 4, -3, 4, 0, 8, -9, 16, -27, 0, 16, -27, 64, -135, 256, 0, 32, -81, 256, -675, 1536, -3125, 0, 64, -243, 1024, -3375, 9216, -21875, 46656, 0, 128, -729, 4096, -16875, 55296, -153125, 373248, -823543, 0, 256, -2187, 16384, -84375, 331776, -1071875, 2985984, -7411887, 16777216
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1-k)^(k-1)*(1+k)^(n-k) for k>0, while T(n,0)=0 by convention.

Examples

			The first rows of the triangle are:
0
0 1
0 2  -1
0 4  -3  4
0 8  -9  16 -27
0 16 -27 64 -135 256

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=0;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k]=(1-k*b)^(k-1)*(1+k*b)^(n-k);););
  return(v);}
  a=seq(100,1)

A244125 Triangle read by rows: terms T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 0, 1, 0, 4, -1, 0, 12, -9, 4, 0, 32, -54, 64, -27, 0, 80, -270, 640, -675, 256, 0, 192, -1215, 5120, -10125, 9216, -3125, 0, 448, -5103, 35840, -118125, 193536, -153125, 46656, 0, 1024, -20412, 229376, -1181250, 3096576, -4287500, 2985984, -823543
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1-k)^(k-1)*(1+k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0 by convention.

Examples

			First rows of the triangle, all summing up to 2^n-1:
1
0 1
0 4  -1
0 12 -9 4
0 32 -54 64 -27
0 80 -270 640 -675 256

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax, b)={my(v, n, k, irow);
  v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
  for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k]=(1-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n,k); ); );
  return(v); }
  a=seq(100, 1)

A244126 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 0, -3, 16, 0, 0, 3, -32, 125, 0, 0, -3, 64, -375, 1296, 0, 0, 3, -128, 1125, -5184, 16807, 0, 0, -3, 256, -3375, 20736, -84035, 262144, 0, 0, 3, -512, 10125, -82944, 420175, -1572864, 4782969, 0, 0, -3, 1024, -30375, 331776, -2100875, 9437184, -33480783, 100000000, 0, 0, 3, -2048, 91125, -1327104, 10504375, -56623104, 234365481, -800000000, 2357947691, 0, 0, -3, 4096, -273375, 5308416, -52521875
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1+k)^(k-1)*(1-k)^(n-k) for k>0, while T(n,0)=0 by convention.

Examples

			The first rows of the triangle are:
0,
0, 1,
0, 0, 3,
0, 0, -3, 16,
0, 0, 3, -32, 125,
0, 0, -3, 64, -375, 1296,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

  • PARI
    seq(nmax, b)={my(v, n, k, irow);
  v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
  for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
    for(k=1, n, v[irow+k]=(1-k*b)^(k-1)*(1+k*b)^(n-k); ); );
  return(v); }
  a=seq(100,-1)
Showing 1-10 of 27 results. Next

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