A088956 -id:A088956 - cp's OEIS frontend

A215652 Exponential Riordan array [exp(x*exp(-x)),x].

1, 1, 1, -1, 2, 1, -2, -3, 3, 1, 9, -8, -6, 4, 1, -4, 45, -20, -10, 5, 1, -95, -24, 135, -40, -15, 6, 1, 414, -665, -84, 315, -70, -21, 7, 1, 49, 3312, -2660, -224, 630, -112, -28, 8, 1, -10088, 441, 14904, -7980, -504, 1134, -168, -36, 9, 1

Offset: 0

Peter Bala, Sep 11 2012

For commuting lower unitriangular matrices A and B we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential and the matrix logarithm Log(A) is defined as sum {n >= 1} (-1)^(n+1)*(A-1)^n/n. Call the present triangle X and let P denote Pascal's triangle A007318. Then X solves the matrix equation X^^P = P. Equivalently, the infinite tower of matrix powers X^^(X^^(X^^(....))) equals P. Note that the infinite tower of powers P^^(P^^(P^^(...))) of the Pascal triangle equals the hyperbinomial array A088956. Thus we might view the present array as the hypobinomial triangle.

			Triangle begins
.n\k.|....0.....1.....2.....3.....4.....5.....6.....7
= = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....1.....1
..2..|...-1.....2.....1
..3..|...-2....-3.....3.....1
..4..|....9....-8....-6.....4.....1
..5..|...-4....45...-20...-10.....5.....1
..6..|..-95...-24...135...-40...-15.....6.....1
..7..|..414..-665...-84...315...-70...-21.....7.....1
...

G. Helms, Pascalmatrix tetrated

Cf. A003506, A003725 (column 0), A007318, A088956.

max = 9; MapIndexed[ Take[#1, #2[[1]]]&, CoefficientList[ Series[ Exp[x*t]*Exp[x*Exp[-x]], {x, 0, max}, {t, 0, max}], {x, t}]*Range[0, max]!, 1] // Flatten (* Jean-François Alcover, Jan 08 2013 *)

T(n,k) = binomial(n,k)*A003725(n-k).
The triangle equals P^^Q, where P is Pascal's triangle and Q is the inverse of P. Column 0 equals A003725.
E.g.f.: exp(x*t)*exp(x*exp(-x)) = 1 + (1 + t)*x + (-1 + 2*t + t^2)*x^2/2! + (-2 - 3*t + 3*t^2 + t^3)*x^3/3! + ....
The infinitesimal generator for this triangle is the generalized exponential Riordan array [x*exp(-x),x], which factors as [x,x]*[exp(-x),x] = A132440*A007318^(-1). The infinitesimal generator begins
..0
..1....0
.-2....2....0
..3...-6....3....0
.-4...12..-12....4....0
This is a signed version of the triangle of denominators from Leibniz's harmonic triangle - see A003506.

A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 50, 24, 6, 1, 432, 200, 48, 8, 1, 4802, 2160, 500, 80, 10, 1, 65536, 28812, 6480, 1000, 120, 12, 1, 1062882, 458752, 100842, 15120, 1750, 168, 14, 1, 20000000, 8503056, 1835008, 268912, 30240, 2800, 224, 16, 1, 428717762, 180000000, 38263752, 5505024, 605052, 54432, 4200, 288, 18, 1

Offset: 0

Paul D. Hanna, Nov 05 2003

Equals the matrix square of A088956 when treated as a lower triangular matrix. The 2nd hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k), where T(n,k) = 2*(n-k+2)^(n-k-1)*C(n,k). Given a table in which the n-th row is the n-th binomial transform of the first row, then the 2nd hyperbinomial transform of any diagonal results in the diagonal located 2 diagonals lower in the table.

			Rows begin:
  {1},
  {2,1},
  {8,4,1},
  {50,24,6,1},
  {432,200,48,8,1},
  {4802,2160,500,80,10,1},
  {65536,28812,6480,1000,120,12,1},
  {1062882,458752,100842,15120,1750,168,14,1},..

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A089461(row sums), A089462(diagonal), A089463, A088956.

Join[{1}, Table[Binomial[n, k]*2*(n - k + 2)^(n - k - 1), {n, 1, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)

for(n=0,10, for(k=0,n, print1(2*(n-k+2)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 18 2017

T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k).
E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2.
Note: (-LambertW(-y)/y)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*y^n/n!.

A089467 Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.

Original entry on oeis.org

1, 2, 8, 52, 478, 5706, 83824, 1461944, 29510268, 676549450, 17361810016, 492999348348, 15345359136232, 519525230896322, 19005788951346240, 747102849650454256, 31404054519248544016, 1405608808807797838866, 66741852193123060505728, 3350816586986433907218500, 177352811048578736727396576

Offset: 0

Paul D. Hanna, Nov 08 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

Cf. A089466, A089468, A088956.

Flatten[{1, Table[Sum[Sum[Binomial[m, j] * Binomial[n, n-m-j] * n^(n-m-j) * (m+j)! / (-2)^j / m!, {j,0,m}], {m,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Oct 11 2020 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!))

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*A089466(k).
a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1)*C(n, k)*A089468(k).
a(n) = Sum_{m=0..n} (Sum_{j=0..m} C(m, j)*C(n, n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!.
a(n) ~ exp(1/2) * n^n. - Vaclav Kotesovec, Oct 11 2020

More terms from Michel Marcus, Jan 12 2025

A089900 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 11, 4, 1, 120, 49, 18, 5, 1, 720, 261, 92, 27, 6, 1, 5040, 1631, 536, 159, 38, 7, 1, 40320, 11743, 3552, 1029, 256, 51, 8, 1, 362880, 95901, 26608, 7353, 1848, 389, 66, 9, 1, 3628800, 876809, 223456, 58095, 14384, 3125, 564, 83, 10, 1

Offset: 0

Paul D. Hanna, Nov 14 2003

Row 1 is A001339, antidiagonal sums form A089902 and the main diagonal is A089901; the next lower diagonal forms {1,4,27,256,..,n^n,..}, which is the hyperbinomial transform (cf. A088956) of the main diagonal.

			Note secondary diagonal: {(n+1)^(n+1)}; rows begin:
1, 2,. 6,. 24,. 120,.. 720,.. 5040,..
1, 3, 11,. 49,. 261,. 1631,. 11743,..
1,_4, 18,. 92,. 536,. 3552,. 26608,..
1, 5,_27, 159, 1029,. 7353,. 58095,..
1, 6, 38,_256, 1848, 14384, 121264,..
1, 7, 51, 389,_3125, 26595, 241015,..
1, 8, 66, 564, 5016,_46656, 456048,..
1, 9, 83, 787, 7701, 78077,_823543,..

Cf. A001339, A088956, A089901, A089902.

t[n_, k_] := (n^(k+2) - Exp[n]*(n-k-1)*Gamma[k+2, n])/(k+1) // Round; Table[t[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)

T(n,k)=if(n<0 || k<0,0,sum(i=0,k,n^(k-i)*binomial(k,i)*(i+1)!))

T(0, k)=(k+1)!, T(n+1, n)=(n+1)^(n+1), T(n, k)=sum_{i=0..k}n^(k-i)*binomial(k, i)*(i+1)!

E.g.f.: 1/((1-y*exp(x))*(1-x)^2). E.g.f. (n-th row): exp(n*x)/(1-x)^2.

A218496 4th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 5, 33, 281, 2993, 38705, 592489, 10516441, 212841889, 4845154913, 122664558905, 3421333467689, 104297273041969, 3451364116327249, 123251578626936841, 4725537745859375705, 193647372258547916609, 8447809104669814884545, 390938955429073736493145

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=4 of A144303.

a:= n-> add(4*(n-j+4)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

E.g.f.: exp(x) * (-LambertW(-x)/x)^4.
a(n) = Sum_{j=0..n} 4 * (n-j+4)^(n-j-1) * C(n,j).
Hyperbinomial transform of A089464.
a(n) ~ 4*exp(4+exp(-1))*n^(n-1). - Vaclav Kotesovec, Aug 16 2013

A218497 5th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 6, 46, 441, 5156, 71801, 1166886, 21756251, 458803176, 10814534541, 282098765426, 8074875680471, 251807768368956, 8501320507058801, 309046115586282726, 12039399243732745851, 500492026353038459216, 22119195334250297991701, 1035767312348853244634586

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=5 of A144303.

a:= n-> add(5*(n-j+5)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[5*(n-j+5)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^5.
a(n) = Sum_{j=0..n} 5 * (n-j+5)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218496.
a(n) ~ 5*exp(5+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218498 6th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 7, 61, 649, 8257, 123217, 2120545, 41484625, 911339617, 22249542241, 598364232529, 17591851634353, 561695417002225, 19366094448215665, 717377453802538753, 28423991158962139873, 1199873992182732076225, 53772852099331738315969, 2550272224743737587911025

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=6 of A144303.

a:= n-> add(6*(n-j+6)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[6*(n-j+6)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^6.
a(n) = A(n,k) = Sum_{j=0..n} 6 * (n-j+6)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218497.
a(n) ~ 6*exp(6+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218499 7th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 8, 78, 911, 12524, 199403, 3624706, 74300269, 1699264792, 42964199279, 1191492782054, 35994106307321, 1177389200637028, 41482632276082915, 1566911405137366450, 63190697224460246477, 2710704012199936430000, 123277690401078017104343, 5925900498827152433216446

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=7 of A144303.

a:= n-> add(7*(n-j+7)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[7*(n-j+7)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^7.
a(n) = Sum_{j=0..n} 7 * (n-j+7)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218498.
a(n) ~ 7*exp(7+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218500 8th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 9, 97, 1233, 18209, 308129, 5901489, 126560849, 3010775745, 78805945665, 2253470828561, 69959985025841, 2345132738183841, 84468280694319713, 3254988169237833585, 133676275015986223569, 5830402582814375609729, 269227430712934320151169

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=8 of A144303.

a:= n-> add(8*(n-j+8)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[8*(n-j+8)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x](-LambertW[-x]/x)^8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 04 2019 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^8.
a(n) = Sum_{j=0..n} 8 * (n-j+8)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218499.
a(n) ~ 8*exp(8+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218501 9th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 10, 118, 1621, 25588, 458605, 9232894, 206835751, 5113191304, 138473150833, 4081818946330, 130223467785619, 4473867764956204, 164772507070721989, 6479598382677480286, 271083794667222927655, 12026359894442420178064, 564099525344446492486105

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=9 of A144303.

a:= n-> add(9*(n-j+9)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[9*(n-j+9)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^9.
a(n) = Sum_{j=0..n} 9 * (n-j+9)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218500.
a(n) ~ 9*exp(9+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

cp's OEIS Frontend

A215652 Exponential Riordan array [exp(x*exp(-x)),x].

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A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

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A089467 Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.

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A089900 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

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A218496 4th iteration of the hyperbinomial transform on the sequence of 1's.

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A218497 5th iteration of the hyperbinomial transform on the sequence of 1's.

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A218498 6th iteration of the hyperbinomial transform on the sequence of 1's.

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A218499 7th iteration of the hyperbinomial transform on the sequence of 1's.

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