A137156 -id:A137156 - cp's OEIS frontend

A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 36, 16, 1, 1, 6, 35, 120, 136, 32, 1, 1, 7, 56, 330, 816, 528, 64, 1, 1, 8, 84, 792, 3876, 5984, 2080, 128, 1, 1, 9, 120, 1716, 15504, 52360, 45760, 8256, 256, 1, 1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896

Offset: 0

Paul D. Hanna, Jan 24 2008

Matrix inverse is A137156.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n-k indistinguishable elements. - Isaac R. Browne, Jun 04 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   4,    1;
  1,  4,  10,    8,     1;
  1,  5,  20,   36,    16,      1;
  1,  6,  35,  120,   136,     32,      1;
  1,  7,  56,  330,   816,    528,     64,      1;
  1,  8,  84,  792,  3876,   5984,   2080,    128,     1;
  1,  9, 120, 1716, 15504,  52360,  45760,   8256,   256,   1;
  1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896, 512, 1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.

Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).

Cf. A092056 (same with reflected rows).

Table[Binomial[2^k+n-k-1,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2017 *)

{T(n,k)=binomial(2^k+n-k-1,n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

{T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(2^k), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A137157 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(22^n).

Original entry on oeis.org

1, 2, 5, 24, 239, 4920, 206727, 17568408, 3003763243, 1030272816360, 707851744149198, 973425618916674288, 2678332881795756783639, 14741522294008025924154864, 162290544340043699103996962253, 3573515596966915773419766367302288, 157376160486791180710467411977740266927

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Equals unsigned column 1 of triangle A137156.

Cf. A137156; A001192, A137158, A137159, A137160.

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+x+x*O(x^n))^(2^(k+1)) ), n)}

a(15)-a(16) from Alois P. Heinz, Jan 04 2023

A137158 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(42^n).

Original entry on oeis.org

1, 4, 22, 228, 4749, 200240, 17034964, 2913479848, 999402129243, 686662003846640, 944294243796543974, 2598186366278914473948, 14300408328085246335179009, 157434326611214704329370130880

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Equals unsigned column 2 of triangle A137156.

Cf. A137156; A001192, A137157, A137159, A137160.

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+x+x*O(x^n))^(2^(k+2)) ), n)}

A137159 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(82^n).

Original entry on oeis.org

1, 8, 92, 1976, 84086, 7173240, 1227862380, 421296930984, 289484024512093, 398106386971472608, 1095381029276651137560, 6028986377761538637043792, 66373632185586959347740452492, 1461497816340787260620205149915824

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Equals unsigned column 3 of triangle A137156.

Cf. A137156; A001192, A137157, A137158, A137160.

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+x+x*O(x^n))^(2^(k+3)) ), n)}

A137160 G.f.: 1 = Sum_{n>=0} a(n)x^n(1-x)/(1+x)^(2^n).

Original entry on oeis.org

1, 2, 4, 12, 64, 664, 13856, 584668, 49751520, 8509625760, 2919099754336, 2005648412219832, 2758163973596156000, 7588978611071894509464, 41769719229784446295570112, 459846172005153447271276789620

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Equals the unsigned row sums of triangle A137156.

Cf. A137156; A001192, A137157, A137158, A137159.

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x)/(1+x+x*O(x^n))^(2^k) ), n)}

A137154 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, n-k); equals the row sums of triangle A137153.

Original entry on oeis.org

1, 2, 4, 9, 24, 79, 331, 1803, 12954, 123983, 1592513, 27604172, 648528166, 20722205191, 903019659239, 53792176322629, 4388683843024734, 491232972054490915, 75545748143323475653, 15984344095578889888206

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Matrix inverse of A137153 is A137156.

Cf. A137153, A137155, A137156.

Table[Sum[Binomial[2^(n-k) + k - 1, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2021 *)

a(n)=sum(k=0,n,binomial(2^k+n-k-1,n-k))

{a(n)=local(A=sum(k=0,n,x^k/(1-x+x*O(x^n))^(2^k)));polcoeff(A,n)} \\ Paul D. Hanna, Sep 15 2009

G.f.: Sum_{n>=0} x^n/(1-x)^(2^n). - Paul D. Hanna, Sep 15 2009

G.f.: Sum_{n>=0} ( (-log(1 - x))^n / n! ) / (1 - 2^n*x). - Paul D. Hanna, Jan 23 2021

cp's OEIS Frontend

A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

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A137157 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2*2^n).

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Extensions

A137158 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^(4*2^n).

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A137159 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^(8*2^n).

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A137160 G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1-x)/(1+x)^(2^n).

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A137154 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, n-k); equals the row sums of triangle A137153.

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Formula

A137157 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(22^n).

A137158 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(42^n).

A137159 G.f.: 1 = Sum_{n>=0} a(n)x^n/(1+x)^(82^n).

A137160 G.f.: 1 = Sum_{n>=0} a(n)x^n(1-x)/(1+x)^(2^n).