A052582 -id:A052582 - cp's OEIS frontend

A042977 Triangle T(n,k) read by rows: coefficients of a polynomial sequence occurring when calculating the n-th derivative of Lambert function W.

1, -2, -1, 9, 8, 2, -64, -79, -36, -6, 625, 974, 622, 192, 24, -7776, -14543, -11758, -5126, -1200, -120, 117649, 255828, 248250, 137512, 45756, 8640, 720, -2097152, -5187775, -5846760, -3892430, -1651480, -445572, -70560, -5040

Offset: 0

Wouter Meeussen

The first derivative of the Lambert W function is given by dW/dz = exp(-W)/(1+W). Further differentiation yields d^2/dz^2(W) = exp(-2*W)*(-2-W)/(1+W)^3, d^3/dz^3(W) = exp(-3*W)*(9+8*W+2*W^2)/(1+W)^5 and, in general, d^n/dz^n(W) = exp(-n*W)*R(n,W)/(1+W)^(2*n-1), where R(n,W) are the row polynomials of this triangle. - Peter Bala, Jul 22 2012

Conjecture: the polynomials have no real roots greater than or equal to -1. This is equivalent to the statement that the derivatives of the 0th branch of the Lambert W function have no real roots greater than -1/e. - Colin Linzer, Jan 29 2025

			Triangle begins:
 n\k |    1    W   W^2   W^3   W^4
==================================
  1  |    1
  2  |   -2   -1
  3  |    9    8     2
  4  |  -64  -79   -36    -6
  5  |  625  974   622   192    24
...
T(5,2) = -4*(-79) - 9*(-36) + 3*(-6) = 622.

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
A. F. Beardon, Winding Numbers, Unwinding Numbers, and the Lambert W Function, Computational Methods and Function Theory, 2021.
George C. Greubel, On Szasz-Mirakyan-Jain Operators preserving exponential functions, arXiv:1805.06968 [math.CA], 2018.
Roy M. Howard, Schröder Based Series for the Lambert W Function, Curtin Univ. (Australia), ResearchGate (2025). See p. 8. See also On Schröder-Type Series Expansions for the Lambert W Function, AppliedMath (2025) Vol. 5, No. 2, Art. No. 66.
G. A. Kalugin and D. J. Jeffrey, Unimodal sequences show that Lambert is Bernstein, C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) pp. 50-56, 2011; arXiv:1011.5940 [math.CA], 2010.
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A013703 (twice row sums), A000444, A000525, A064781, A064785, A064782.
First column A000169, main diagonal A000142, first subdiagonal A052582.
Cf. A054589.

# After Vladimir Kruchinin, for 0 <= m <= n:
T := (n, m) -> add(add((-1)^(k+n)*binomial(j,k)*binomial(2*n+1,m-j)*(k+n+1)^(n+j), k=0..j)/j!, j=0..m): seq(seq(T(n, k), k=0..n), n=0..7); # Peter Luschny, Feb 23 2018

Table[ Simplify[ (Evaluate[ D[ ProductLog[ z ], {z, n} ] ] /. ProductLog[ z ]->W)*z^n/W^n (1+W)^(2n-1) ], {n, 12} ] // TableForm
Flatten[ Table[ CoefficientList[ Simplify[ (Evaluate[D[ProductLog[z], {z, n}]] /. ProductLog[z] -> W) z^n / W^n (1 + W)^(2 n - 1)], W], {n, 8}]] (* Michael Somos, Jun 07 2012 *)
T[ n_, k_] := If[ n < 1 || k < 0, 0, Coefficient[ Simplify[(Evaluate[D[ProductLog[z], {z, n}]] /. ProductLog[z] -> W) z^n / W^n (1 + W)^(2 n - 1)], W, k]] (* Michael Somos, Jun 07 2012 *)

B(n):=(if n=1 then 1/(1+x)*exp(-x) else -n!*sum((sum((-1)^(m-j)*binomial(m,j)*sum((j^(n-i)*binomial(j,i)*x^(m-i))/(n-i)!,i,0,n),j,1,m))*B(m)/m!,m,1,n-1)/(1+x)^n);
a(n):=B(n)*(1+x)^(2*n-1);
/* Vladimir Kruchinin, Apr 07 2011 */

a(n):=if n=1 then 1 else (n-1)!*(sum((binomial(n+k-1, n-1)*sum(binomial(k, j)*(x+1)^(n-j-1)*sum(binomial(j, l)*(-1)^(l)*sum((l^(n+j-i-1)*binomial(l, i)*x^(j-i))/(n+j-i-1)!, i, 0, l), l, 1, j), j, 1, k)), k, 1, n-1));
T(n, k):=coeff(ratsimp(a(n)), x, k);
for n: 1 thru 12 do print(makelist(T(n, k), k, 0, n-1));
/* Vladimir Kruchinin, Oct 09 2012 */
T(n,m):=sum(binomial(2*n+1,m-j)*sum(((n+k+1)^(n+j)*(-1)^(n+k))/((j-k)!*k!),k,0,j),j,0,m); /* Vladimir Kruchinin, Feb 20 2018 */

E.g.f.: (LambertW(exp(x)*(x+y*(1+x)^2))-x)/(1+x). - Vladeta Jovovic, Nov 19 2003
a(n) = B(n)*(1+x)^(2*n-1), where B(1) = 1/(1+x), and for n>=2, B(n) = -(n!/(1+x)^n)*Sum_{m=1..n-1} (B(m)/m!)*Sum_{j=1..m} (-1)^(m-j)*binomial(m,j)*Sum_{i=0..n} j^(n-i)*binomial(j,i)*x^(m-i)/(n-i)!. - Vladimir Kruchinin, Apr 07 2011
Recurrence equation: T(n+1,k) = -n*T(n,k-1) - (3*n-k-1)*T(n,k) + (k+1)*T(n,k+1). - Peter Bala, Jul 22 2012
T(n,m) = Sum_{j=0..m} C(2*n+1,m-j)*(Sum_{k=0..j} (n+k+1)^(n+j)*(-1)^(n+k)/((j-k)!*k!)). - Vladimir Kruchinin, Feb 20 2018

A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

Original entry on oeis.org

1, 1, -2, 2, -8, 7, 6, -36, 63, -34, 24, -192, 504, -544, 209, 120, -1200, 4200, -6800, 5225, -1546, 720, -8640, 37800, -81600, 94050, -55656, 13327, 5040, -70560, 370440, -999600, 1536150, -1363572, 653023, -130922, 40320, -645120, 3951360, -12794880

Offset: 0

Roger L. Bagula, Oct 04 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A021009(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
       1;
       1,    -2;
       2,    -8,     7;
       6,   -36,    63,    -34;
      24,  -192,   504,   -544,   209;
     120, -1200,  4200,  -6800,  5225,  -1546;
     720, -8640, 37800, -81600, 94050, -55656, 13327;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 13 2018

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, p. 782.
Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=0..100 of triangle, flattened; offset corrected by _Georg Fischer_, Jan 31 2019)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Laguerre Polynomial

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

M := (n,x) -> n!*subs(x=(x/(1-x)),orthopoly[L](n,x))*(1-x)^n:
seq(print(seq(coeff(simplify(M(n,x)),x,k),k=0..n)),n=0..6); # Peter Luschny, Jan 05 2015

w = Table[n!*CoefficientList[LaguerreL[n, x], x], {n, 0, 10}];
v = Table[CoefficientList[Sum[w[[n + 1]][[m + 1]]*x^ m*(1 - x)^(n - m), {m, 0, n}], x], {n, 0, 10}]; Flatten[v]

create_list(ratcoef(n!*(1 - x)^n*laguerre(n, x/(1 - x)), x, k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Oct 13 2018 */

row(n) = Vecrev(n!*(1-x)^n*pollaguerre(n, 0, x/(1 - x))); \\ Michel Marcus, Feb 06 2021

T(n, k) = [x^k] (n!*L_n(x)*(1 - x)^n) with L_n(x) the Laguerre polynomial after substituting x by x/(1 - x). - Peter Luschny, Jan 05 2015
From Franck Maminirina Ramaharo, Oct 13 2018: (Start)
G.f.: exp(-x*y/(1 - (1 - x)*y))/(1 - (1 - x)*y).
T(n,1) = A000142(n).
T(n,2) = -A052582(n).
T(n,n) = A002720(n). (End)

Edited by N. J. A. Sloane, Jun 12 2007

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018

A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 4, 0, 5, 6, 10, 2, 1, 21, 36, 42, 12, 9, 0, 117, 226, 219, 104, 47, 6, 1, 792, 1568, 1472, 800, 328, 64, 16, 0, 6205, 12360, 11596, 6652, 2658, 688, 148, 12, 1, 55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25, 0, 543597, 1085560, 1030649, 614420, 255830, 77732, 17750, 2876, 365, 20, 1

Offset: 0

Alois P. Heinz, Jan 23 2019

			T(4,0) = 5: 1234, 1432, 3214, 3412, 4231.
T(4,1) = 6: 2431, 3241, 3421, 4132, 4213, 4312.
T(4,2) = 10: 1243, 1324, 1342, 1423, 2134, 2314, 2413, 3124, 3142, 4321.
T(4,3) = 2: 2341, 4123.
T(4,4) = 1: 2143.
Triangle T(n,k) begins:
      1;
      1,      0;
      1,      0,      1;
      2,      0,      4,     0;
      5,      6,     10,     2,     1;
     21,     36,     42,    12,     9,    0;
    117,    226,    219,   104,    47,    6,    1;
    792,   1568,   1472,   800,   328,   64,   16,   0;
   6205,  12360,  11596,  6652,  2658,  688,  148,  12,  1;
  55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25,  0;
  ...

Alois P. Heinz, Rows n = 0..24, flattened

Column k=0 gives A078480.
Row sums give A000142.
Main diagonal gives A059841.
Cf. A008290, A008291, A052582, A082033, A323671.

b:= proc(s) option remember; expand((n-> `if`(n=0, 1, add(
     `if`(abs(n-j)=1, x, 1)*b(s minus {j}), j=s)))(nops(s)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n})):
seq(T(n), n=0..12);

b[s_] := b[s] = Expand[With[{n = Length[s]}, If[n==0, 1, Sum[
     If[Abs[n-j]==1, x, 1]*b[s~Complement~{j}], {j, s}]]]];
T[n_] := PadRight[CoefficientList[b[Range[n]], x], n+1];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A052582(n-1) for n > 0.

Sum_{k=0..n} (k+1) * T(n,k) = A082033(n-1) for n > 0.

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

Original entry on oeis.org

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Emeric Deutsch, Apr 06 2008

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

T(n,k) = 2*(n-k-1)*(n-2)!.
T(n,0) = 2(n-1)! = A052849(n-1).
T(n,1) = A052582(n-2).
T(n,2) = A052609(n-2).
T(n,3) = 12*A005990(n-3).
T(n,4) = 48*A061206(n-5).
T(n,n-2) = 2(n-2)! (A052849).
Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).

A108861 Numbers k that divide the sum of the digits of 2^k * k!.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 27, 81, 126, 159, 205, 252, 254, 267, 285, 675, 1053, 1086, 1125, 1146, 2007, 5088, 5382, 5448, 14652, 23401, 23574, 24009, 41004, 66789, 67482, 111480, 866538, 1447875, 2413152, 2414019, 2417828, 2421360, 4045482, 6713982

Offset: 1

Ryan Propper, Jul 11 2005

Next term after 5448 is greater than 10000.
a(34) > 10^6. - D. S. McNeil, Mar 03 2009
a(39) > 2.54 * 10^6, if it exists. - Kevin P. Thompson, Oct 20 2021
a(41) > 7*10^6, if it exists. - Kevin P. Thompson, Dec 08 2021

			9 is a term because the sum of the digits of 2^9 * 9! = 185794560 is 45 which is divisible by 9.

Cf. A000165, A007953, A052582.

Do[If[Mod[Plus @@ IntegerDigits[2^n * n! ], n] == 0, Print[n]], {n, 1, 10000}]
Select[Range[6714000],Mod[Total[IntegerDigits[2^# #!]],#]==0&] (* Harvey P. Dale, Jul 11 2023 *)

isok(k) = !(sumdigits(2^k * k!) % k); \\ Michel Marcus, Oct 20 2021

from itertools import islice
def A108861(): # generator of terms
    k, k2, kf = 1, 2, 1
    while True:
        c = sum(int(d) for d in str(k2*kf))
        if not c % k: yield k
        k += 1
        k2 *= 2
        kf *= k
A108861_list = list(islice(A108861(),10)) # Chai Wah Wu, Oct 26 2021

a(25)-a(33) from D. S. McNeil, Mar 03 2009
a(34)-a(38) from Kevin P. Thompson, Oct 20 2021
a(39)-a(40) from Kevin P. Thompson, Dec 08 2021

A066237 First differences give A052849.

Original entry on oeis.org

1, 3, 7, 19, 67, 307, 1747, 11827, 92467, 818227, 8075827, 87909427, 1045912627, 13499954227, 187856536627, 2803205272627, 44648785048627, 756023641240627, 13560771052696627, 256850971870360627, 5122654988223640627

Offset: 1

Markus Sullivan (markus(AT)o-reading.co.uk), Dec 19 2001

Partial sums of A098558. - Sébastien Desbordes, Dec 18 2023

Sébastien Desbordes, Showing why it is the partial sums of A098558

Cf. A003422, A052849, A098558, A052582.

RecurrenceTable[{a[0]==-1,a[1]==1,a[n]==n*a[n-1]-(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Dec 10 2013 *)

From Vladeta Jovovic, Dec 20 2001: (Start)
a(n) = n*a(n-1) - (n-1)*a(n-2), a(0)=-1, a(1)=1.
a(n) = 2*A003422(n) - 1. (End)

More terms from Jason Earls, Jan 13 2002

A120928 Number of "ups" and "downs" in the permutations of [n] if either a previous counted "up" ("down") or a "void" precedes an "up" ("down") which then will be counted also.

Original entry on oeis.org

2, 8, 44, 280, 2040, 16800, 154560, 1572480, 17539200, 212889600, 2794176000, 39437798400, 595718323200, 9589612032000, 163895187456000, 2964061900800000, 56554301067264000, 1135354270482432000, 23923536413736960000, 527939735774330880000

Offset: 2

Thomas Wieder, Jul 16 2006

An "up" ("down") is a neighboring pair of elements e_i, e_j of [n] with e_i < e_j (e_i > e_j). A "void" is a missing preceding pair, i.e., the start of [n]. We discuss two examples for [n=4]. In the permutation [3, 1, 2, 4] "void" precedes the pair 3,1 and consequently a "down" is counted. No "up" which has been counted precedes the "ups" 1,2 and 2,4 so they are not counted. In [3, 4, 1, 2] the "up" 3,4 is counted and so is the next "up" 1,2 but the down 4,1 has no preceding "down" registered and is therefore not counted.

			[1, 2, 3, 4], "ups"=3, "downs"=0;
[1, 2, 4, 3], "ups"=2, "downs"=0;
[1, 3, 2, 4], "ups"=2, "downs"=0;
[1, 3, 4, 2], "ups"=2, "downs"=0;
[1, 4, 2, 3], "ups"=2, "downs"=0;
[1, 4, 3, 2], "ups"=1, "downs"=0;
[2, 1, 3, 4], "ups"=0, "downs"=1;
[2, 1, 4, 3], "ups"=0, "downs"=2;
[2, 3, 1, 4], "ups"=2, "downs"=0;
[2, 3, 4, 1], "ups"=2, "downs"=0;
[2, 4, 1, 3], "ups"=2, "downs"=0;
[2, 4, 3, 1], "ups"=1, "downs"=0;
[3, 1, 2, 4], "ups"=0, "downs"=1;
[3, 1, 4, 2], "ups"=0, "downs"=2;
[3, 2, 1, 4], "ups"=0, "downs"=2;
[3, 2, 4, 1], "ups"=0, "downs"=2;
[3, 4, 1, 2], "ups"=2, "downs"=0;
[3, 4, 2, 1], "ups"=1, "downs"=0;
[4, 1, 2, 3], "ups"=0, "downs"=1;
[4, 1, 3, 2], "ups"=0, "downs"=2;
[4, 2, 1, 3], "ups"=0, "downs"=2;
[4, 2, 3, 1], "ups"=0, "downs"=2;
[4, 3, 1, 2], "ups"=0, "downs"=2;
[4, 3, 2, 1], "ups"=0, "downs"=3.

Alois P. Heinz, Table of n, a(n) for n = 2..400

Cf. A028399, A052582, A062119, A097971.

a:= n-> ceil(n!*(3*n-1)/6):
seq(a(n), n=2..30); # Alois P. Heinz, Apr 21 2012

E.g.f.: -(6+6*x^2-4*x^3+x^4)/(-3+12*x-18*x^2+12*x^3-3*x^4). - Thomas Wieder, May 02 2009

a(2) = 2, a(n) = n! * (3*n - 1) / 6 for n > 2. - Jon E. Schoenfield, Apr 18 2010

4 more terms from R. J. Mathar, Aug 25 2008

More terms from Alois P. Heinz, Apr 21 2012

A060694 A triangle related to rooted trees.

Original entry on oeis.org

1, 2, 1, 10, 8, 2, 82, 86, 36, 6, 938, 1202, 668, 192, 24, 13778, 20772, 14118, 5452, 1200, 120, 247210, 427828, 341122, 161688, 48312, 8640, 720, 5240338, 10228458, 9325398, 5184902, 1909920, 467784, 70560, 5040, 128149802, 278346286

Offset: 1

F. Chapoton, Apr 20 2001

The rows sum to A006963, the alternating sum is A000311, the right column is A000142, the left column is related to A032188 (twice); the second-to-right column is A052582

			{1}, {2,1}, {10,8,2}, {82,86,36,6}, {938,1202,668,192,24}

Cf. A006963, A000311, A000142, A032188, A052582.

E.g.f. given by the Maple expression RootOf(-exp(_Z*x*t)+x*t*exp(_Z*x*t)+y*t*exp(-_Z+_Z*x*t)-y*t^2*x*exp(-_Z+_Z*x*t)+1-t+t*exp(-_Z+_Z*x*t)-x*t*exp(-_Z+_Z*x*t));

More terms from Vladeta Jovovic, Apr 21 2001

cp's OEIS Frontend

A042977 Triangle T(n,k) read by rows: coefficients of a polynomial sequence occurring when calculating the n-th derivative of Lambert function W.

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A123202 Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

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A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

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A108861 Numbers k that divide the sum of the digits of 2^k * k!.

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A066237 First differences give A052849.

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A120928 Number of "ups" and "downs" in the permutations of [n] if either a previous counted "up" ("down") or a "void" precedes an "up" ("down") which then will be counted also.

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A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.