A089946 -id:A089946 - cp's OEIS frontend

A089945 Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

1, 3, 15, 112, 1125, 14256, 218491, 3932160, 81310473, 1900000000, 49516901511, 1424099377152, 44804009850925, 1530735634132992, 56439656982421875, 2233785415175766016, 94459960699823921169, 4250383588380798812160, 202774313738037680879743

Offset: 0

Paul D. Hanna, Nov 23 2003

nonn

a(n) is the number of labeled trees on n+1 nodes with a designated node or edge. - Geoffrey Critzer, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A089944, A089946, A245348.

[(2*n+1)*(n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

nn = 30; T[z_] = -LambertW[-z]; Drop[Range[0, nn]! CoefficientList[Series[T[z] + T[z]^2/2, {z, 0, nn}], z], 1] (* Geoffrey Critzer, Mar 25 2017 *)
Table[(2 n + 1) (n + 1)^(n - 1), {n, 0, 18}] (* Michael De Vlieger, Mar 25 2017 *)

PARI
```
a(n)=if(n<0,0,(2*n+1)*(n+1)^(n-1))
```

a(n) = (2*n+1)*(n+1)^(n-1).

E.g.f.: (-LambertW(-x)/x)*(1-LambertW(-x))/(1+LambertW(-x)).

A276911 E.g.f. A(x) satisfies: A(A( xexp(-x) )) = xexp(x).

Original entry on oeis.org

1, 2, 6, 28, 180, 1446, 13888, 156472, 2034000, 29724490, 476806176, 8502508884, 174802753216, 3768345692398, 63300353418240, 1386349221087856, 149879079531401472, 5097575010920072850, -780487993325688128000, -32524149870689487270260, 10927977097616993825596416, 490896441869732669067535414, -213936255246865273137807851520, -10450262329586550037066790750808, 6047981224337998054714885264691200

Offset: 1

Paul D. Hanna, Sep 22 2016

sign

Former name was "Inverse of e.g.f. A(x) equals its conjugate, where A(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1)." - Paul D. Hanna, Sep 06 2018

			E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + ...
such that A(A( x*exp(-x) )) = x*exp(x).
RELATED SERIES.
Let F(x) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ a(n)*i^(n-1)*x^n/n! +...
then
(a) Series_Reversion( F(x) ) = conjugate( F(x) ).
(b) F(x) = G(x)*exp(i*G(x)) where G(x) is the e.g.f. of A276910:
(c) G(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! +...+ A276910(n)*x^n/n! +...
where
G( F(x) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...

Paul D. Hanna, Table of n, a(n) for n = 1..301

Cf. A276910, A276912.

{a(n) = my(V=[1],A=x,G=x); for(i=1,n\2+1, V = concat(V,[0,0]); G = sum(m=1,#V,V[m]*x^m/m!) +x*O(x^#V);
A = G*exp(I*G); V[#V] = -(#V)!/2 * polcoeff( subst( A, x, conj(A) ),#V) ); n!*(-I)^(n-1)*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x). - Paul D. Hanna, Sep 06 2018
E.g.f. A(x) satisfies: A(-A(-x)) = x. - Paul D. Hanna, Sep 06 2018
Inverse of F(x) equals its conjugate, where F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1).
Let G(x) be the e.g.f. of A276910, then F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! satisfies:
(1) F(x) = G(x) * exp(i*G(x)).
(2) G( F(x) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.
E.g.f. A(x) satisfies: A(A(x)) is e.g.f. of A089946 with offset 1. - Alexander Burstein, Jan 15 2022

Name replaced with simpler formula by Paul D. Hanna, Sep 06 2018

A291203 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0

Offset: 0

Alois P. Heinz, Aug 20 2017

Positive elements in column t=1 give A034855.
Elements in rows h=0 give A023531.
Elements in rows h=1 give A059297.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.

			n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272
-----+-------------------+---------+------------------+--------
0 0  : 1                 :         :                  : 1
-----+-------------------+---------+------------------+--------
1 0  : 0   1             :      1  :   .              :
1 1  : 0                 :         :   1              : 1
-----+-------------------+---------+------------------+--------
2 0  : 0   0   1         :      1  :   .   .          :
2 1  : 0   2             :      2  :   .              :
2 2  : 0                 :         :   2   1          : 3
-----+-------------------+---------+------------------+--------
3 0  : 0   0   0  1      :      1  :   .   .   .      :
3 1  : 0   3   6         :      9  :   .   .          :
3 2  : 0   6             :      6  :   .              :
3 3  : 0                 :         :   9   6   1      : 16
-----+-------------------+---------+------------------+--------
4 0  : 0   0   0  0  1   :      1  :   .   .   .  .   :
4 1  : 0   4  24 12      :     40  :   .   .   .      :
4 2  : 0  36  24         :     60  :   .   .          :
4 3  : 0  24             :     24  :   .              :
4 4  : 0                 :         :  64  48  12  1   : 125
-----+-------------------+---------+------------------+--------
5 0  : 0   0   0  0  0 1 :      1  :   .   .   .  . . :
5 1  : 0   5  80 90 20   :    195  :   .   .   .  .   :
5 2  : 0 200 300 60      :    560  :   .   .   .      :
5 3  : 0 300 120         :    420  :   .   .          :
5 4  : 0 120             :    120  :   .              :
5 5  : 0                 :         : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*j*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
     Binomial[n-1, j-1]*j*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
F(n,0,0) = A000007(n).

A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

Original entry on oeis.org

1, 4, 2, 24, 15, 9, 200, 136, 100, 64, 2160, 1535, 1215, 945, 625, 28812, 21036, 17286, 14406, 11526, 7776, 458752, 341103, 286671, 247296, 211456, 172081, 117649, 8503056, 6405904, 5464712, 4811528, 4251528, 3691528, 3038344, 2097152, 180000000, 136953279, 118078911, 105372819, 94921875, 85078125, 74627181, 61921089, 43046721

Offset: 1

Rui Duarte, Jan 22 2018

T(n,k) is the number of pairs (f,i) such that f is a parking function and f(i) = k.

			Triangle begins:
====================================================================
n\k|       1       2       3       4       5       6       7       8
---|----------------------------------------------------------------
1  |       1
2  |       4       2
3  |      24      15       9
4  |     200     136     100      64
5  |    2160    1535    1215     945     625
6  |   28812   21036   17286   14406   11526    7776
7  |  458752  341103  286671  247296  211456  172081  117649
8  | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
  ...

Cf. A000169, A000272, A089946, A298592, A298594, A298597.

Table[n Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

T(n,k) = n*Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
T(n,k) = n*A298592(n,k).
T(n,k) = n*Sum_{j=k..n} A298594(n,j).
T(n,k) = Sum_{j=k..n} A298597(n,j).
Sum_{k=1..n} T(n,k) = n*A000272(n+1).
T(n+1,1) = A089946(n), T(n,n) = A000169(n). - Andrey Zabolotskiy, Feb 21 2018

A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704, 99, 11, 1

Offset: 0

Paul D. Hanna, Nov 23 2003

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.

			Rows begin:
  {1, 2, 3, 4, 5, 6, 7,..},
  {1, 3, 8, 20, 48, 112, 256,..},
  {1, 4, 15, 54, 189, 648, 2187,..},
  {1, 5, 24, 112, 512, 2304, 10240,..},
  {1, 6, 35, 200, 1125, 6250, 34375,..},
  {1, 7, 48, 324, 2160, 14256, 93312,..},
  {1, 8, 63, 490, 3773, 28812, 218491,..},..

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals).

Cf. A089945, A089946.
Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.
Columns : A000012, A000027, A005563.

A089944[n_, k_] := (k + n + 1)*(n + 1)^(k - 1);
Table[A089944[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

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

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

E.g.f.: (1+x)*exp(x)/(1-y*exp(x)).

A259334 Triangle read by rows: T(n,k) = k(n-1)!n^(n-k-1)/(n-k)!, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 16, 24, 18, 6, 125, 200, 180, 96, 24, 1296, 2160, 2160, 1440, 600, 120, 16807, 28812, 30870, 23520, 12600, 4320, 720, 262144, 458752, 516096, 430080, 268800, 120960, 35280, 5040, 4782969, 8503056, 9920232, 8817984, 6123600, 3265920, 1270080, 322560, 40320

Offset: 1

N. J. A. Sloane, Jun 25 2015

			Triangle begins:
     1;
     1,    1;
     3,    4,    2;
    16,   24,   18,    6;
   125,  200,  180,   96,   24;
  1296, 2160, 2160, 1440,  600,  120;
  ...

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Diagonals include A000272, A089946, A000142.

Cf. A000435.

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(k*(n-1)!*n^(n-k-1)/(n-k)!, ", ");); print(););} \\ Michel Marcus, Jun 26 2015

A000435(n) = Sum_{k=0..n-1} k*T(n,k). - David desJardins, Jan 22 2017

More terms from Michel Marcus, Jun 26 2015

A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 9, 12, 3, 64, 96, 36, 4, 625, 1000, 450, 80, 5, 7776, 12960, 6480, 1440, 150, 6, 117649, 201684, 108045, 27440, 3675, 252, 7, 2097152, 3670016, 2064384, 573440, 89600, 8064, 392, 8, 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9

Offset: 1

Geoffrey Critzer, May 08 2013

Row sums = 2n*(n+1)^(n-2) = A089946(offset).
The average number of trees in each forest approaches 5/2 as n gets large.
The rows give the coefficients of the derivatives of the Abel polynomials. - Peter Luschny, Feb 22 2025

			    T(2,1)=2                  T(2,2)=2
  ...1'...   ...2'...   ...1'..2...   ...1..2'...
  ...| ...   ...| ...   ...........   ...........
  ...2 ...   ...1 ...   ...........   ...........
The root node is on top.  The ' indicates the tree which has been specially designated.
Triangle starts:
  [1]        1;
  [2]        2,        2;
  [3]        9,       12,        3;
  [4]       64,       96,       36,        4;
  [5]      625,     1000,      450,       80,       5;
  [6]     7776,    12960,     6480,     1440,     150,      6;
  [7]   117649,   201684,   108045,    27440,    3675,    252,     7;
  [8]  2097152,  3670016,  2064384,   573440,   89600,   8064,   392,   8;
  [9] 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9;

Cf. A061356, A089946 (row sums), A000169, A137452.

Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1, 8}] // Grid

T(n, k) = binomial(n-1, k-1)*n^(n-k)*k = A061356(n, k)*k(offset).

E.g.f.: y*A(x)*exp(y*A(x)) where A(x) is e.g.f. for A000169.

cp's OEIS Frontend

A089945 Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

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A276911 E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x).

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A291203 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

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A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

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A259334 Triangle read by rows: T(n,k) = k*(n-1)!*n^(n-k-1)/(n-k)!, 1 <= k <= n.

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A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.

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A276911 E.g.f. A(x) satisfies: A(A( xexp(-x) )) = xexp(x).

A259334 Triangle read by rows: T(n,k) = k(n-1)!n^(n-k-1)/(n-k)!, 1 <= k <= n.