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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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)

A244127 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, 0, 3, 0, 0, -9, 16, 0, 0, 18, -128, 125, 0, 0, -30, 640, -1875, 1296, 0, 0, 45, -2560, 16875, -31104, 16807, 0, 0, -63, 8960, -118125, 435456, -588245, 262144, 0, 0, 84, -28672, 708750, -4644864, 11764900, -12582912, 4782969
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:
0,
0, 1,
0, 0, 3,
0, 0, -9, 16,
0, 0, 18, -128, 125,
0, 0, -30, 640, -1875, 1296,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, 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)

A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

Original entry on oeis.org

0, 1, 0, 1, -2, 0, 1, -4, 9, 0, 1, -8, 27, -64, 0, 1, -16, 81, -256, 625, 0, 1, -32, 243, -1024, 3125, -7776, 0, 1, -64, 729, -4096, 15625, -46656, 117649, 0, 1, -128, 2187, -16384, 78125, -279936, 823543, -2097152, 0, 1, -256, 6561, -65536, 390625, -1679616, 5764801, -16777216, 43046721
Offset: 1

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(-k)^(k-1)*k^(n-k) for k>0, while T(n,0)=0 by convention. The flattened triangle start with row 1, coefficient T(1,0).
Resembles A076014, but with added powers of 0, and with sign-alternating columns.

Examples

			The first rows of the triangle (starting at n=1):
0, 1,
0, 1, -2,
0, 1, -4, 9,
0, 1, -8, 27, -64,
0, 1, -16, 81, -256, 625,
0, 1, -32, 243, -1024, 3125, -7776,

Links

Crossrefs

Cf. A076014, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, 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)/2-1);
for(n=1,nmax,irow=n*(n+1)/2;v[irow]=0;
  for(k=1,n,v[irow+k]=(-1)^(k-1)*(k*b)^(n-1);););
return(v);}
a=seq(100,1);

A244129 Triangle read by rows: terms of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 1, 0, 2, -2, 0, 3, -12, 9, 0, 4, -48, 108, -64, 0, 5, -160, 810, -1280, 625, 0, 6, -480, 4860, -15360, 18750, -7776, 0, 7, -1344, 25515, -143360, 328125, -326592, 117649, 0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152
Offset: 1

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

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

Examples

			First rows of the triangle, starting at row n=1. All rows sum up to 0, except the first one whose sum is 1:
0, 1;
0, 2, -2;
0, 3, -12, 9;
0, 4, -48, 108, -64;
0, 5, -160, 810, -1280, 625;
0, 6, -480, 4860, -15360, 18750, -7776;
0, 7, -1344, 25515, -143360, 328125, -326592, 117649;
0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152; ...
From _Paul D. Hanna_, Sep 13 2017: (Start)
E.g.f.: A(x,y) = y*x + (-2*y^2 + 2*y)*x^2/2! + (9*y^3 - 12*y^2 + 3*y)*x^3/3! + (-64*y^4 + 108*y^3 - 48*y^2 + 4*y)*x^4/4! + (625*y^5 - 1280*y^4 + 810*y^3 - 160*y^2 + 5*y)*x^5/5! + (-7776*y^6 + 18750*y^5 - 15360*y^4 + 4860*y^3 - 480*y^2 + 6*y)*x^6/6! + (117649*y^7 - 326592*y^6 + 328125*y^5 -  143360*y^4 + 25515*y^3 - 1344*y^2 + 7*y)*x^7/7! +...
such that A(x,y) * exp( A(x,y) ) = y*x*exp(x). (End)

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, 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)/2-1);
for(n=1, nmax, irow=n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k]=(-1)^(k-1)*(k*b)^(n-1)*binomial(n,k); ); );
return(v); }
a=seq(100, 1);

Formula

E.g.f. A(x,y) satisfies: A(x,y) * exp( A(x,y) ) = y*x*exp(x). - Paul D. Hanna, Sep 13 2017

A244130 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum_{k=0..n} T(n,k)*binomial(n,k).

Original entry on oeis.org

0, 0, 1, 0, 2, -2, 0, 4, -6, 9, 0, 8, -18, 36, -64, 0, 16, -54, 144, -320, 625, 0, 32, -162, 576, -1600, 3750, -7776, 0, 64, -486, 2304, -8000, 22500, -54432, 117649, 0, 128, -1458, 9216, -40000, 135000, -381024, 941192, -2097152, 0, 256, -4374, 36864, -200000, 810000, -2667168, 7529536, -18874368, 43046721
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k) = (-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, -2,
0, 4, -6, 9,
0, 8, -18, 36, -64,
0, 16, -54, 144, -320, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, 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]=(-k*b)^(k-1)*(1+k*b)^(n-k);););
return(v);}
a=seq(100,1);

A244131 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 0, 1, 0, 4, -2, 0, 12, -18, 9, 0, 32, -108, 144, -64, 0, 80, -540, 1440, -1600, 625, 0, 192, -2430, 11520, -24000, 22500, -7776, 0, 448, -10206, 80640, -280000, 472500, -381024, 117649, 0, 1024, -40824, 516096, -2800000, 7560000, -10668672, 7529536, -2097152
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(-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 n:
0,
0, 1,
0, 4, -2,
0, 12, -18, 9,
0, 32, -108, 144, -64,
0, 80, -540, 1440, -1600, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, 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]=(-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n,k); ); );
return(v); }
a=seq(100,1);

A244132 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k)*binomial(n,k).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, -2, 9, 0, 0, 2, -18, 64, 0, 0, -2, 36, -192, 625, 0, 0, 2, -72, 576, -2500, 7776, 0, 0, -2, 144, -1728, 10000, -38880, 117649, 0, 0, 2, -288, 5184, -40000, 194400, -705894, 2097152, 0, 0, -2, 576, -15552, 160000, -972000, 4235364, -14680064, 43046721
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(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, 2,
0, 0, -2, 9,
0, 0, 2, -18, 64,
0, 0, -2, 36, -192, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, 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]=(-k*b)^(k-1)*(1+k*b)^(n-k); ); );
return(v); }
a=seq(100,-1);

A244133 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, -6, 9, 0, 0, 12, -72, 64, 0, 0, -20, 360, -960, 625, 0, 0, 30, -1440, 8640, -15000, 7776, 0, 0, -42, 5040, -60480, 210000, -272160, 117649, 0, 0, 56, -16128, 362880, -2240000, 5443200, -5647152, 2097152, 0, 0, -72, 48384, -1959552, 20160000, -81648000, 152473104, -132120576, 43046721
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(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 n:
0,
0, 1,
0, 0, 2,
0, 0, -6, 9,
0, 0, 12, -72, 64,
0, 0, -20, 360, -960, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, 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]=(-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n, k); ); );
return(v); }
a=seq(100,-1);

A244134 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, 3, -2, 0, 16, -10, 9, 0, 125, -72, 63, -64, 0, 1296, -686, 576, -576, 625, 0, 16807, -8192, 6561, -6400, 6875, -7776, 0, 262144, -118098, 90000, -85184, 90000, -101088, 117649, 0, 4782969, -2000000, 1449459, -1327104, 1373125, -1524096, 1764735, -2097152
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(-k)^(k-1)*(n+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, 3, -2,
0, 16, -10, 9,
0, 125, -72, 63, -64,
0, 1296, -686, 576, -576, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, 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]=(-k*b)^(k-1)*(n+k*b)^(n-k);););
return(v);}
a=seq(100,1);

A244135 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, 6, -2, 0, 48, -30, 9, 0, 500, -432, 252, -64, 0, 6480, -6860, 5760, -2880, 625, 0, 100842, -122880, 131220, -96000, 41250, -7776, 0, 1835008, -2480058, 3150000, -2981440, 1890000, -707616, 117649, 0, 38263752, -56000000, 81169704, -92897280, 76895000, -42674688, 14117880, -2097152
Offset: 0

Views

Author

Stanislav Sykora, Jun 22 2014

Keywords

Comments

T(n,k)=(-k)^(k-1)*(n+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, 6, -2,
0, 48, -30, 9,
0, 500, -432, 252, -64,
0, 6480, -6860, 5760, -2880, 625,

Links

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, 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]=(-k*b)^(k-1)*(n+k*b)^(n-k)*binomial(n,k); ); );
return(v); }
a=seq(100, 1);
Previous Showing 11-20 of 27 results. Next

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