A080108 -id:A080108 - cp's OEIS frontend

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1

Offset: 0

Paul Curtz, Jan 08 2009

From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
      1;
     -1,      1;
      3,     -4,     1;
    -16,     24,    -9,     1;
    125,   -200,    90,   -16,   1;
  -1296,   2160, -1080,   240, -25,   1;
  16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021

			With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1        \      /1        \
|1 1     ||0 1       ||0 1      |      |1  1      |
|1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
|1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
|...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
|...     ||...       ||...      |      |          |
- _Peter Bala_, Jan 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A080108, A152818, A059297, A145033.

/* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016

T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)

T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)

A174493 a(n) = coefficient of x^n/(n-1)! in the 3-fold iteration of x*exp(x).

Original entry on oeis.org

1, 3, 15, 102, 861, 8598, 98547, 1270160, 18138601, 283754826, 4818884319, 88186786020, 1728395865021, 36091833338174, 799408841413051, 18708996086926272, 461095012437724881, 11931573394008790290

Offset: 1

Paul D. Hanna, Apr 17 2010

nonn

			E.g.f.: x + 3*x^2 + 15*x^3/2! + 102*x^4/3! + 861*x^5/4! +...

Cf. A174480, A080108, A174494, A174495, A174496.

{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n)));for(i=1,3,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}

{a(n)=sum(k=0,n-1,binomial(n-1,k)*sum(j=0,n-1-k,binomial(n-1-k,j)*(k+1)^j*(k+1+j)^(n-1-k-j)))}

a(n) = Sum_{k=0..n, j=0..n-k} C(n,k)*C(n-k,j)*(k+1)^j*(k+1+j)^(n-k-j).

O.g.f.: Sum_{n>=1} A080108(n)*x^n/(1-n*x)^n, where A080108(n) = [x^n/(n-1)! ] E(E(x)) and E(x) = x*exp(x).

A174494 a(n) = coefficient of x^n/(n-1)! in the 4-fold iteration of x*exp(x).

Original entry on oeis.org

1, 4, 28, 274, 3400, 50734, 880312, 17357736, 382463824, 9298086490, 246914949376, 7104423326356, 220000621675912, 7290852811359654, 257332393857067720, 9632914084301343304, 381050245422453157408

Offset: 1

Paul D. Hanna, Apr 17 2010

nonn

			E.g.f.: x + 4*x^2 + 28*x^3/2! + 274*x^4/3! + 3400*x^5/4! +...

Cf. A174480, A080108, A174493, A174495, A174496.

{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n)));for(i=1,4,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}

{a(n)=sum(k=0,n-1,binomial(n-1,k)*sum(j=0,n-1-k,binomial(n-1-k,j)*(k+1)^j*sum(i=0,n-1-k-j,binomial(n-1-k-j,i)*(k+1+j)^i*(k+1+j+i)^(n-1-k-j-i))))}

O.g.f.: Sum_{n>=1} A174493(n)*x^n/(1-n*x)^n, where A174493(n) = [x^n/(n-1)! ] E(E(E(x))) and E(x) = x*exp(x).
a(n)=Sum_{k=0..n-1, j=0..n-1-k, i=0..n-1-k-j} C(n-1,k)*C(n-1-k,j)*C(n-1-k-j,i)*(k+1)^j*(k+1+j)^i*(k+1+j+i)^(n-1-k-j-i).
E.g.f. equals the 2-fold iteration of the e.g.f. of A080108.

A174495 a(n) = coefficient of x^n/(n-1)! in the 5-fold iteration of x*exp(x).

Original entry on oeis.org

1, 5, 45, 575, 9425, 187455, 4367245, 116322645, 3479863345, 115353325835, 4192244804645, 165607074622665, 7060695856372105, 322973775761169135, 15770136907303728205, 818373668098974428885, 44963322539225628107105

Offset: 1

Paul D. Hanna, Apr 17 2010

nonn

			E.g.f.: x + 5*x^2 + 45*x^3/2! + 575*x^4/3! + 9425*x^5/4! +...

Cf. A174480, A080108, A174493, A174494, A174496.

{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n))); for(i=1,5,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}

O.g.f.: Sum_{n>=1} A174494(n)*x^n/(1-n*x)^n, where A174494(n) = [x^n/(n-1)! ] E(E(E(E(x)))) and E(x) = x*exp(x).

A355472 Expansion of Sum_{k>=0} (x/(1 - k^3 * x))^k.

Original entry on oeis.org

1, 1, 2, 18, 275, 6680, 258897, 13646776, 959706169, 88651586048, 10272048320897, 1462972094910224, 253355867842243905, 52387780870782231424, 12745274175326359046785, 3615579524073585972982544, 1184928928181459098548941633, 444427677344332049739011858432

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..235

Cf. A080108, A355463, A355471.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^3*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(3*(n-k))*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k^(3*(n-k)) * binomial(n-1,k-1) for n > 0.

A356811 a(n) = Sum_{k=0..n} (kn+1)^(n-k) binomial(n,k).

Original entry on oeis.org

1, 2, 8, 71, 1040, 22457, 676000, 26861977, 1347932416, 82873789793, 6114540967424, 532596023373713, 53990083205042176, 6289985311473281329, 833180470332123750400, 124356049859476364116193, 20754548375601491155681280, 3847574240184742568296430273

Offset: 0

Seiichi Manyama, Aug 29 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..274

Cf. A080108, A240165, A245834.

Cf. A356806, A356814, A356817.

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

a(n) = n! * [x^n] exp( x * (exp(n * x) + 1) ).

a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k+1)*x)^(k+1).

A360708 Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 14, 42, 136, 479, 1825, 7433, 32053, 145608, 695081, 3479117, 18209842, 99373513, 563920590, 3320674902, 20255823092, 127799984935, 832807892861, 5597481205009, 38753768384761, 276057156622776, 2021100095469577, 15193591060371577

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..630

Cf. A080108, A360709.

Cf. A000248, A360699.

Join[{1},Table[Sum[Binomial[n-k-1,k-1] * k^(n-2*k), {k,0,n/2}], {n,1,40}]] (* Vaclav Kotesovec, Feb 20 2023 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x^2/(1-k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n\2, k^(n-2*k)*binomial(n-k-1, k-1)));

a(n) = Sum_{k=1..floor(n/2)} k^(n-2*k) * binomial(n-k-1,k-1) for n > 0.

A360709 Expansion of Sum_{k>=0} (x^3 / (1 - k*x))^k.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 5, 13, 34, 90, 247, 720, 2256, 7568, 26814, 98982, 377541, 1484254, 6021789, 25271173, 109850447, 494355359, 2298362532, 11008133629, 54175202125, 273460921605, 1414449612648, 7494262602464, 40669492399396

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..692

Cf. A080108, A360708.

Cf. A078012, A360707.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x^3/(1-k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n\3, k^(n-3*k)*binomial(n-2*k-1, k-1)));

a(n) = Sum_{k=1..floor(n/3)} k^(n-3*k) * binomial(n-2*k-1,k-1) for n > 0.

A098697 Euler-Seidel matrix T(k,n) with start sequence A000248, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 6, 4, 3, 23, 17, 13, 10, 104, 81, 64, 51, 41, 537, 433, 352, 288, 237, 196, 3100, 2563, 2130, 1778, 1490, 1253, 1057, 19693, 16593, 14030, 11900, 10122, 8632, 7379, 6322, 136064, 116371, 99778, 85748, 73848, 63726, 55094, 47715, 41393

Offset: 0

Ralf Stephan, Sep 23 2004

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.

			1,1,3,10,41,196,1057,
2,4,13,51,237,1253,7379,
6,17,64,288,1490,8632,55094,
23,81,352,1778,10122,63726,437810,
104,433,2130,11900,73848,501536,3687056,

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

First column is A080108, main diagonal is in A098698.

a248[0] = 1; a248[n_] := Sum[Binomial[n, k]*(n - k)^k, {k, 0, n}];
T[0, n_] := T[0, n] = a248[n];
T[k_, n_] := T[k, n] = T[k - 1, n] + T[k - 1, n + 1];
Table[T[k - n, n], {k, 0, 9}, {n, 0, k}] // Flatten (* Jean-François Alcover, Nov 08 2017 *)

Recurrence: T(0, n) = A000248(n), T(k, n) = T(k-1, n) + T(k-1, n+1).

A174496 a(n) = coefficient of x^n/(n-1)! in the 6-fold iteration of x*exp(x).

Original entry on oeis.org

1, 6, 66, 1041, 21216, 527631, 15441636, 518651881, 19630068656, 825581830491, 38159948599956, 1921319136589221, 104603652465885096, 6120324106269585751, 382829011514506048556, 25484466375276284094561

Offset: 1

Paul D. Hanna, Apr 17 2010

nonn

			E.g.f.: x + 6*x^2 + 66*x^3/2! + 1041*x^4/3! + 21216*x^5/4! +...

Cf. A174480, A080108, A174493, A174494, A174495.

{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n))); for(i=1,6,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}

O.g.f.: Sum_{n>=1} A174495(n)*x^n/(1-n*x)^n, where A174495(n) = [x^n/(n-1)! ] E(E(E(E(E(x))))) and E(x) = x*exp(x).
E.g.f. equals the 2-fold iteration of the e.g.f. of A174493.
E.g.f. equals the 3-fold iteration of the e.g.f. of A080108.

cp's OEIS Frontend

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

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A174493 a(n) = coefficient of x^n/(n-1)! in the 3-fold iteration of x*exp(x).

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A174494 a(n) = coefficient of x^n/(n-1)! in the 4-fold iteration of x*exp(x).

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A174495 a(n) = coefficient of x^n/(n-1)! in the 5-fold iteration of x*exp(x).

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A355472 Expansion of Sum_{k>=0} (x/(1 - k^3 * x))^k.

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A356811 a(n) = Sum_{k=0..n} (k*n+1)^(n-k) * binomial(n,k).

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A360708 Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.

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Mathematica

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A360709 Expansion of Sum_{k>=0} (x^3 / (1 - k*x))^k.

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A356811 a(n) = Sum_{k=0..n} (kn+1)^(n-k) binomial(n,k).