A159964 -id:A159964 - cp's OEIS frontend

A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

1, 1, 1, 1, 1, 5, 1, 20, 3, 1, 84, 35, 1, 409, 295, 15, 1, 2365, 2359, 315, 1, 16064, 19670, 4480, 105, 1, 125664, 177078, 56672, 3465, 1, 1112073, 1738326, 703430, 74025, 945, 1, 10976173, 18607446, 8941790, 1346345, 45045, 1, 119481284, 216400569, 118685336

Offset: 0

Vladeta Jovovic, May 03 2008

			Triangle (n,k) begins:
  1;
  1;
  1,    1;
  1,    5;
  1,   20,    3;
  1,   84,   35;
  1,  409,  295,  15;
  1, 2365, 2359, 315;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021. See Theorem 2. p. 5.
FindStat - Combinatorial Statistic Finder, The number of nontrivial cycles of a permutation pi in its cycle decomposition
Bin Han, Jianxi Mao, and Jiang Zeng, Equidistributions around special kinds of descents and excedances, arXiv:2103.13092 [math.CO], 2021, see page 2.

Columns k=0-10 give: A000012, A006231, A289950, A289951, A289952, A289953, A289954, A289955, A289956, A289957, A289958.
Row sums give A000142.
Cf. A000276, A000483, A001147, A001705, A051577, A124324, A159964.

egf:= proc(k::nonnegint) option remember; x-> exp(x)* ((-x-ln(1-x))^k)/k! end; T:= (n,k)-> coeff(series(egf(k)(x), x=0, n+1), x, n) *n!; seq(seq(T(n,k), k=0..n/2), n=0..30); # Alois P. Heinz, Aug 14 2008
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      `if`(i>1, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 25 2016
# third Maple program:
T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
      `if`(n=0, 1, add(T(n-i, k-`if`(i>1, 1, 0))*
       mul(n-j, j=1..i-1), i=1..n)))
    end:
seq(seq(T(n,k), k=0..n/2), n=0..15);  # Alois P. Heinz, Jul 16 2017

max = 12; egf = Exp[x*(1-y)]/(1-x)^y; s = Series[egf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; t[0, 0] = t[1, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

E.g.f.: exp(x*(1-y))/(1-x)^y. Binomial transform of triangle A008306. exp(x)*((-x-log(1-x))^k)/k! is e.g.f. of k-th column.
From Alois P. Heinz, Jul 13 2017: (Start)
T(2n,n) = A001147(n).
T(2n+1,n) = A051577(n) = (2*n+3)!!/3 = A001147(n+2)/3. (End)
From Alois P. Heinz, Aug 17 2023: (Start)
Sum_{k=0..floor(n/2)} k * T(n,k) = A001705(n-1) for n>=1.
Sum_{k=0..floor(n/2)} (-1)^k * T(n,k) = A159964(n-1) for n>=1. (End)

A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, -2, 1, 4, -5, 1, -8, 20, -9, 1, 16, -78, 59, -14, 1, -32, 324, -360, 135, -20, 1, 64, -1520, 2254, -1165, 265, -27, 1, -128, 8336, -15232, 9954, -3045, 469, -35, 1, 256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1, -512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54

Offset: 0

Roger L. Bagula, Apr 08 2008

			{1},
{-2, 1},
{4, -5, 1},
{-8, 20, -9, 1},
{16, -78,59, -14, 1},
{-32, 324, -360, 135, -20, 1},
{64, -1520, 2254, -1165, 265, -27, 1},
{-128, 8336, -15232, 9954, -3045, 469, -35, 1},
{256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1},
{-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1},
{1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1}

M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133.
Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial.

Cf. A046716, A092553, A094816, A159964.

R := proc(n) add((-1)^k*binomial(n,k)* k!*2^(n-k)*binomial(-x, k), k=0..n);
expand(%) end: p := n -> seq((-1)^(n-k)*coeff(R(n), x, k), k=0..n):
seq(p(n), n = 0..9);
# Or:
egf := exp(-2*t)*(1+t)^x: ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 27 2019

Ca[x, 0] = 1; Ca[x, 1] = -2 + x;
Ca[x_, n_] := Ca[x, n] = (x - n - 1) Ca[x, n - 1] - 2 (n - 1) Ca[x, n - 2];
Table[CoefficientList[Ca[x, n], x], {n, 0, 9}] // Flatten
(* The unsigned row polynomials (see Peter Bala's comment) are: *)
R[n_] := HypergeometricU[-n, 1 - n - x, 2];
Table[R[n], {n, 0, 6}] (* Peter Luschny, Oct 27 2019 *)

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(-2*t)*(1+t)^x.
With p(0, x) = 1 and p(1, x) = x - 2 the polynomials obey the recurrence
p(n, x) = (x - n - 1)*p(n-1, x) - 2*(n - 1)*p(n-2, x).
Row sums are (-2)^n*(n-1) = (-1)^n*A159964(n-1).
From Peter Bala, Oct 23 2019: (Start)
The unsigned row polynomials are
   R(n,x) = Sum_{k=0..n} (-1)^k*binomial(n, k)*k!*2^(n-k)*binomial(-x, k).
They occur in series acceleration formulas for the constant
   1/e^2 = n!*2^n*Sum_{k >= 0}(-2)^k/(k!*R(n,k)*R(n,k+1)) = 0.1353 35283 23661 ... (cf. A092553, A046716, A094816).
(End)
R(n, x) = KummerU(-n, 1 - n - x, 2). - Peter Luschny, Oct 27 2019

Edited by Peter Luschny, Oct 27 2019

A124791 Row sums of number triangle A124790.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 29, 73, 181, 465, 1205, 3171, 8425, 22597, 61073, 166195, 454949, 1251985, 3461573, 9611191, 26787377, 74916661, 210178457, 591347989, 1668172841, 4717282753, 13369522249, 37970114703, 108045430901

Offset: 0

Paul Barry, Nov 07 2006

nonn

Row sums of a generalized Motzkin triangle.
Apparently the Motzkin transform of A105811, after the sign of A105811(1) is negated. - R. J. Mathar, Dec 11 2008
Hankel transform is A159964. - Paul Barry, Apr 28 2009

Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 11.

Cf. A105811, A124790, A159964.

Conjecture: (n+1)*a(n) +(-n+2)*a(n-1) +(-5*n+7)*a(n-2) +3*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2014

A172160 a(0)=1. a(n) = 2^(n-2)*(5-n), for n>0.

Original entry on oeis.org

1, 2, 3, 4, 4, 0, -16, -64, -192, -512, -1280, -3072, -7168, -16384, -36864, -81920, -180224, -393216, -851968, -1835008, -3932160, -8388608, -17825792, -37748736, -79691776, -167772160, -352321536, -738197504, -1543503872, -3221225472, -6710886400

Offset: 0

Paul Curtz, Jan 27 2010

The inverse binomial transform is 1,1,0,0,-1,-1,-2,-2,-3,-3 = essentially A168050 or the negative of A004526.

			G.f. = 1 + 2*x + 3*x^2 + 4*x^3 + 4*x^4 - 16*x^6 - 64*x^7 + ... - _Michael Somos_, Apr 22 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A004526, A045891, A168050.

Cf. A000079, A001787, A059165, A159964, A131577.

Table[2^(n-2)*(5-n) -(1/4)*Boole[n==0], {n,0,40}] (* G. C. Greubel, Apr 21 2022 *)

[2^(n-2)*(5-n) -(1/4)*bool(n==0) for n in (1..40)] # G. C. Greubel, Apr 21 2022

a(n+1) - 2*a(n) = -A131577(n).
a(n) + A001787(n-1) = A000079(n+1).
a(n+5) = -A059165(n) = 4*A159964(n+1).
G.f.: (1 - 2*x - x^2)/(1-2*x)^2. - R. J. Mathar, Feb 11 2010
a(n) = 4*a(n-1) - 4*a(n-2), n>2.
E.g.f.: (1/4)*((5-2*x)*exp(2*x) - 1). - G. C. Greubel, Apr 21 2022
a(n) = 4^n*A045891(1-n) if n>1. - Michael Somos, Apr 22 2022

Definition replaced with closed form by R. J. Mathar, Feb 11 2010

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

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A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

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A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A124791 Row sums of number triangle A124790.

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A172160 a(0)=1. a(n) = 2^(n-2)*(5-n), for n>0.

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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