proposed

approved

editing

proposed

Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j,=0,..n} A051179(j), read by rows.

reviewed

editing

proposed

reviewed

editing

proposed

A product triangle based on A051179:Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product[_{j,0,n} A051179(m),{m,0,n}]; t(n,k)=c(n)/(c(k)*c(n-k)j), read by rows.

1, 1, 1, 1, 5, 1, 1, 85, 85, 1, 1, 21845, 371365, 21845, 1, 1, 1431655765, 6254904037285, 6254904037285, 1431655765, 1, 1, 6148914691236517205, 1760625833240390967011987365, 452480839142780478522080752805, 1760625833240390967011987365, 6148914691236517205, 1

Row sums are:

{1, 2, 7, 172, 415057, 12512671386102, 456002090821559089838577761947, ...}.

G. C. Greubel, <a href="/A173475/b173475.txt">Rows n = 0..10 of the triangle, flattened</a>

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product[_{j,0,n} A051179(mj),{m,0,n}];.

t(n,k)=c(n)/(c(k)*c(n-k))

{1},

Triangle begins as:

1;

{ 1, 1},;

{ 1, 5, 1},;

{ 1, 85, 85, 1},;

{ 1, 21845, 371365, 21845, 1},;

{ 1, 1431655765, 6254904037285, 6254904037285, 1431655765, 1},;

{1, 6148914691236517205, 1760625833240390967011987365, 452480839142780478522080752805, 1760625833240390967011987365, 6148914691236517205, 1},

{1, 113427455640312821154458202477256070485, 139491149675259572522122379028593954502678535399079038885, 2349450689400744057326259929080019233034205095833506843651404965, 2349450689400744057326259929080019233034205095833506843651404965, 139491149675259572522122379028593954502678535399079038885, 113427455640312821154458202477256070485, 1}

c[n_] := Product[2^(2^mj) - 1, {m, j, 0, n}];

tT[n_, k_] := c[n]/(c[k]*c[n - k]);

Table[Table[tT[n, k], {n, 0, 8}, {k, 0, n}], {n, 0, 10}]//Flatten

(Sage)

@CachedFunction

def c(n): return product( 2^(2^j) -1 for j in (0..n) )

def T(n, k): return c(n)/(c(k)*c(n-k))

flatten([[T(n, k) for k in (0..n)] for n in (0..8)]) # G. C. Greubel, Apr 26 2021

Cf. A051179.

nonn,tabl,unedless

Edited by G. C. Greubel, Apr 26 2021

approved

editing

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 19 2010

A product triangle based on A051179:c(n)=Product[A051179(m),{m,0,n}]; t(n,k)=c(n)/(c(k)*c(n-k))

1, 1, 1, 1, 5, 1, 1, 85, 85, 1, 1, 21845, 371365, 21845, 1, 1, 1431655765, 6254904037285, 6254904037285, 1431655765, 1, 1, 6148914691236517205, 1760625833240390967011987365, 452480839142780478522080752805

0,5

Row sums are:

{1, 2, 7, 172, 415057, 12512671386102, 456002090821559089838577761947, ...}.

c(n)=Product[A051179(m),{m,0,n}];

t(n,k)=c(n)/(c(k)*c(n-k))

{1},

{1, 1},

{1, 5, 1},

{1, 85, 85, 1},

{1, 21845, 371365, 21845, 1},

{1, 1431655765, 6254904037285, 6254904037285, 1431655765, 1},

{1, 6148914691236517205, 1760625833240390967011987365, 452480839142780478522080752805, 1760625833240390967011987365, 6148914691236517205, 1},

{1, 113427455640312821154458202477256070485, 139491149675259572522122379028593954502678535399079038885, 2349450689400744057326259929080019233034205095833506843651404965, 2349450689400744057326259929080019233034205095833506843651404965, 139491149675259572522122379028593954502678535399079038885, 113427455640312821154458202477256070485, 1}

c[n_] = Product[2^(2^m) - 1, {m, 0, n}];

t[n_, k_] = c[n]/(c[k]*c[n - k]);

Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]

Cf. A051179

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 19 2010

approved