A244142 -id:A244142 - cp's OEIS frontend

A244136 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).

1, 0, 1, 0, 1, 2, 0, 4, 2, 9, 0, 27, 8, 9, 64, 0, 256, 54, 36, 64, 625, 0, 3125, 512, 243, 256, 625, 7776, 0, 46656, 6250, 2304, 1728, 2500, 7776, 117649, 0, 823543, 93312, 28125, 16384, 16875, 31104, 117649, 2097152, 0, 16777216, 1647086, 419904, 200000, 160000, 209952, 470596, 2097152, 43046721

Offset: 0

Stanislav Sykora, Jun 22 2014

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

			The first rows of the triangle are:
1,
0, 1,
0, 1, 2,
0, 4, 2, 9,
0, 27, 8, 9, 64,
0, 256, 54, 36, 64, 625,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(13), with b=-1.

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

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);

A244137 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, 2, 2, 0, 12, 6, 9, 0, 108, 48, 36, 64, 0, 1280, 540, 360, 320, 625, 0, 18750, 7680, 4860, 3840, 3750, 7776, 0, 326592, 131250, 80640, 60480, 52500, 54432, 117649, 0, 6588344, 2612736, 1575000, 1146880, 945000, 870912, 941192, 2097152

Offset: 0

Stanislav Sykora, Jun 22 2014

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

There are many binomial decompositions of n^n, some with all terms positive like this one (see A243203). However, for every n, the terms corresponding to k=1..n in this one are exceptionally similar in value (at least on log scale).

			First rows of the triangle, all summing up to n^n:
1,
0, 1,
0, 2, 2,
0, 12, 6, 9,
0, 108, 48, 36, 64,
0, 1280, 540, 360, 320, 625,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(13), with b=-1.

Cf. A243203, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244138, A244139, A244140, A244141, A244142, A244143.

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);

A244138 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n(n-1) as Sum(k=0..n)T(n,k)binomial(n,k).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jun 22 2014

T(n,k)=k*(1-k)^(k-2)*k^(n-k) for k>1, while T(n,0)=T(n,1)=0 by convention.

			The first rows of the triangle are:
0,
0, 0,
0, 0, 2,
0, 0, 4, -6,
0, 0, 8, -18, 36,
0, 0, 16, -54, 144, -320,
0, 0, 32, -162, 576, -1600, 3750,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(19), with a=1.

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244139, A244140, A244141, A244142, A244143.

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

A244139 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(n-1) as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 12, -6, 0, 0, 48, -72, 36, 0, 0, 160, -540, 720, -320, 0, 0, 480, -3240, 8640, -9600, 3750, 0, 0, 1344, -17010, 80640, -168000, 157500, -54432, 0, 0, 3584, -81648, 645120, -2240000, 3780000, -3048192, 941192, 0, 0, 9216, -367416, 4644864, -25200000, 68040000, -96018048, 67765824, -18874368

Offset: 0

Stanislav Sykora, Jun 22 2014

T(n,k)=k*(1-k)^(k-2)*k^(n-k)*binomial(n,k) for k>1, while T(n,0)=T(n,1)=0 by convention.

			First rows of the triangle, all summing up to n*(n-1):
0,
0, 0,
0, 0, 2,
0, 0, 12, -6,
0, 0, 48, -72, 36,
0, 0, 160, -540, 720, -320,
0, 0, 480, -3240, 8640, -9600, 3750,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(19), with a=1.

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244140, A244141, A244142, A244143.

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

A244140 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n(-1)^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, 0, -3, 0, 0, 0, -3, 16, 0, 0, 0, -3, 32, -135, 0, 0, 0, -3, 64, -405, 1536, 0, 0, 0, -3, 128, -1215, 6144, -21875, 0, 0, 0, -3, 256, -3645, 24576, -109375, 373248, 0, 0, 0, -3, 512, -10935, 98304, -546875, 2239488, -7411887, 0, 0, 0, -3, 1024, -32805, 393216, -2734375, 13436928, -51883209, 167772160

Offset: 0

Stanislav Sykora, Jun 23 2014

T(n,k)=(-1)^k*k*(k-2)^(n-2) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) by convention.

			The first rows of the triangle are:
0,
0, -1,
0, 0, 2,
0, 0, 0, -3,
0, 0, 0, -3, 16,
0, 0, 0, -3, 32, -135,
0, 0, 0, -3, 64, -405, 1536,
0, 0, 0, -3, 128, -1215, 6144, -21875,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(21), with a=2, b=1.

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244141, A244142, A244143.

seq(nmax)={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;if(n==1,v[irow+1]=-1,v[irow+1]=0);
for(k=2,n,v[irow+k]=(-1)^k*k*(k-2)^(n-2);););
return(v);}
a=seq(100);

A244141 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k).

Original entry on oeis.org

0, 0, -1, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, -12, 16, 0, 0, 0, -30, 160, -135, 0, 0, 0, -60, 960, -2430, 1536, 0, 0, 0, -105, 4480, -25515, 43008, -21875, 0, 0, 0, -168, 17920, -204120, 688128, -875000, 373248, 0, 0, 0, -252, 64512, -1377810, 8257536, -19687500, 20155392, -7411887

Offset: 0

Stanislav Sykora, Jun 23 2014

T(n,k)=(-1)^k*k*(k-2)^(n-2)*binomial(n,k) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) by convention.

			First rows of the triangle, all summing up to n*(-1)^n:
0,
0, -1,
0, 0, 2,
0, 0, 0, -3,
0, 0, 0, -12, 16,
0, 0, 0, -30, 160, -135,
0, 0, 0, -60, 960, -2430, 1536,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(21), with a=2, b=1.

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244142, A244143.

seq(nmax)={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; if(n==1, v[irow+1]=-1, v[irow+1]=0);
for(k=2, n, v[irow+k]=(-1)^k*k*(k-2)^(n-2)*binomial(n,k); ); );
return(v); }
a=seq(100);

A244143 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, 18, -15, 0, 0, 108, -300, 196, 0, 0, 540, -3750, 6860, -3645, 0, 0, 2430, -37500, 144060, -196830, 87846, 0, 0, 10206, -328125, 2352980, -6200145, 6764142, -2599051, 0, 0, 40824, -2625000, 32941720, -148803480, 297622248, -270301304, 91125000

Offset: 0

Stanislav Sykora, Jun 23 2014

T(n,k)=(-1)^k*k*(2*k-1)^(n-2)*binomial(n,k) for k>1, while T(n,0)=0 and T(1,1)=0^(n-1) by convention.

			First rows of the triangle, all summing up to n:
0,
0, 1,
0, 0, 2,
0, 0, 18, -15,
0, 0, 108, -300, 196,
0, 0, 540, -3750, 6860, -3645,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(21), with a=1, b=2.

Cf. A244116, 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.

seq(nmax)={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; if(n==1, v[irow+1]=1, v[irow+1]=0);
for(k=2, n, v[irow+k]=(-1)^k*k*(2*k-1)^(n-2)*binomial(n,k); ); );
return(v); }
a=seq(100);

cp's OEIS Frontend

A244136 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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244137 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244138 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n*(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244139 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(n-1) as Sum(k=0..n)T(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244140 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k)*binomial(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244141 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244143 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A244138 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n(n-1) as Sum(k=0..n)T(n,k)binomial(n,k).

A244140 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n(-1)^n as Sum(k=0..n)T(n,k)binomial(n,k).