A003149 -id:A003149 - cp's OEIS frontend

A084938 Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 24, 16, 9, 4, 1, 0, 120, 64, 31, 14, 5, 1, 0, 720, 312, 126, 52, 20, 6, 1, 0, 5040, 1812, 606, 217, 80, 27, 7, 1, 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1, 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1

Offset: 0

Philippe Deléham, Jul 16 2003; corrections Dec 17 2008, Dec 20 2008, Feb 05 2009

Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0,...] = A110654 DELTA A000007.
In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/(1-(r_2*x+s_2*x*y)/(1-(r_3*x+s_3*x*y)/(1-...(continued fraction). See also the Formula section below.
T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n >= 1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan, Jul 20 2005
T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan, Jul 20 2005
Riordan array (1,x*g(x)) where g(x) is the g.f. of the factorials (n!). - Paul Barry, Sep 25 2008
Modulo 2, this sequence becomes A106344.
T(n,k) is the number of permutations of {1,2,...,n} having k cycles such that the elements of each cycle of the permutation form an interval. - Ran Pan, Nov 11 2016
The convolution triangle of the factorial numbers. - Peter Luschny, Oct 09 2022

			From _Paul Barry_, Sep 25 2008: (Start)
Triangle [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,0,0,0,...] begins
  1;
  0,      1;
  0,      1,     1;
  0,      2,     2,     1;
  0,      6,     5,     3,    1;
  0,     24,    16,     9,    4,    1;
  0,    120,    64,    31,   14,    5,   1;
  0,    720,   312,   126,   52,   20,   6,   1;
  0,   5040,  1812,   606,  217,   80,  27,   7,  1;
  0,  40320, 12288,  3428, 1040,  345, 116,  35,  8, 1;
  0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1. (End)
From _Paul Barry_, May 14 2009: (Start)
The production matrix is
  0,   1;
  0,   1,  1;
  0,   1,  1, 1;
  0,   2,  1, 1, 1;
  0,   7,  2, 1, 1, 1;
  0,  34,  7, 2, 1, 1, 1;
  0, 206, 34, 7, 2, 1, 1, 1;
which is based on A075834. (End)

T. D. Noe, Rows n = 0..100 of triangle, flattened
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), Article 09.7.6.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
David Callan, A combinatorial interpretation of the eigensequence for composition, arXiv:math/0507169 [math.CO], 2005.
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
H. Fuks and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
Peter Luschny, Transformations of integer sequences.
R. J. Mathar, Properties of Deleham's Delta Transformation: OEIS A084938

Cf. A003149, A051295 (row sums), A052186, A090238,
Cf. A287899.
Columns: A000007, A000142, A003149, A090595, A090319.
Diagonals: A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.

function T(n,k) // T = A084938
  if k lt 0 or k gt n then return 0;
  elif n eq 0 or k eq n then return 1;
  elif k eq 0 then return 0;
  else return (&+[Factorial(j)*T(n-j-1,k-1): j in [0..n-1]]);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 10 2022

DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);
for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0,k] := 1; od: for i from 0 to n do P[i,-1] := 0; od:
for i from 1 to n do for k from 0 to n do P[i,k] := sort(expand(P[i,k-1] + q[k]*P[i-1,k+1])); od: od:
for i from 0 to n do t1 := P[i,0]; for j from 0 to i do T[i,j] := coeff(coeff(t1,x,i-j),y,j); od: lprint( seq(T[i,j],j=0..i) ); od: end;
# To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i),i= 0..40)]; s := [seq(s4(i),i=0..40)]; DELTA(r,s,20);
# Uses function PMatrix from A357368.
PMatrix(10, n -> factorial(n - 1)); # Peter Luschny, Oct 09 2022

a[0, 0] = 1; a[n_, k_] := a[n, k] = Sum[j! a[n - j - 1, k - 1], {j, 0, n - 1}]; Flatten[Table[a[i, j], {i, 0, 10}, {j, 0, i}]] (* T. D. Noe, Feb 22 2012 *)
DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[Floor[Range[10]/2], Prepend[Table[0, {10}], 1], 10] (* Jean-François Alcover, Sep 12 2013, after Philippe Deléham *)

def delehamdelta(R, S) :
    L = min(len(R), len(S)) + 1
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [Rk + x*Sk for Rk, Sk in zip(R, S)]
    C = [ring(0)] + [ring(1) for i in range(L)]
    for k in (1..L) :
        for n in range(k-1,0,-1) :
            C[n] = C[n-1] + C[n+1]*A[n-1]
        yield list(C[1])
def A084938_triangle(n) :
    for row in delehamdelta([(i+1)//2 for i in (0..n)], [0^i for i in (0..n)]):
        print(row)
A084938_triangle(10) # Peter Luschny, Jan 28 2012

The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:
Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0. Then P(n, k) is a homogeneous polynomial in x and y of degree n and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).
T(n, n) = 1.
T(k+1, k) = A001477(k).
T(k+2, k) = A000096(k).
T(n+1, 1) = A000142(n).
T(n+2, 2) = A003149(n).
T(n+3, 3) = A090595(n).
T(n+4, 4) = A090319(n).
T(m+n, m) = Sum_{k=0..n} A090238(n, k)*binomial(m, k).
G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.
For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.
T(n,k) = Sum_{j>=0} A075834(j)*T(n-1,k+j-1).
T(2n,n) = A287899(n). - Alois P. Heinz, Jun 02 2017
From G. C. Greubel, Nov 10 2022: (Start)
Sum_{k=0..n} T(n, k) = A051295(n).
Sum_{k=0..n} (-1)^k*T(n, k) = [n=0] - A052186(n-1)*[n>0]. (End)

Name edited by Derek Orr, May 01 2015

A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11

Offset: 1

N. J. A. Sloane

Array 1/Beta(n,m) read by antidiagonals. - Michael Somos, Feb 05 2004
a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - Ross La Haye, Feb 13 2004
a(n,k) = total size of all of the elements of the family of k-size subsets of an n-element set. For example, a 2-element set, say, {1,2}, has 3 families of k-size subsets: one with 1 0-size element, one with 2 1-size elements and one with 1 2-size element; respectively, {{}}, {{1},{2}}, {{1,2}}. - Ross La Haye, Dec 31 2006
Second slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for which the first slice is Pascal's triangle (slice read by antidiagonals). - Thomas Wieder, Aug 06 2006
Triangle, read by rows, given by [2,-1/2,1/2,0,0,0,0,0,0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
This sequence * [1/1, 1/2, 1/3, ...] = (1, 3, 7, 15, 31, ...). - Gary W. Adamson, Nov 14 2007
n-th row = coefficients of first derivative of corresponding Pascal's triangle row. Example: x^4 + 4x^3 + 6x^2 + 4x + 1 becomes (4, 12, 12, 4). - Gary W. Adamson, Dec 27 2007
From Paul Curtz, Jun 03 2011: (Start)
Consider
   1     1/2   1/3    1/4    1/5
  -1/2  -1/6  -1/12  -1/20  -1/30
   1/3   1/12  1/30   1/60   1/105
  -1/4  -1/20 -1/60  -1/140 -1/280
   1/5   1/30  1/105  1/280  1/630
This is an autosequence (the inverse binomial transform is the sequence signed) of the second kind: the main diagonal is 2 times the first upper diagonal.
Note that 2, 12, 60, ... = A005430(n+1), Apery numbers = 2*A002457(n). (End)
From Louis Conover (for the 9th grade G1c mathematics class at the Chengdu Confucius International School), Mar 02 2015: (Start)
The i-th order differences of n^-1 appear in the (i+1)th row.
  1,    1/2,   1/3,   1/4,    1/5,    1/6,    1/7,     1/8, ...
  1/2,  1/6,  1/12,  1/20,   1/30,   1/42,   1/56,    1/72, ...
  1/3, 1/12,  1/30,  1/60,  1/105,  1/168,  1/252,   1/360, ...
  1/4, 1/20,  1/60, 1/140,  1/280,  1/504,  1/840,  1/1320, ...
  1/5, 1/30, 1/105, 1/280,  1/630, 1/1260, 1/2310,  1/3960, ...
  1/6, 1/42, 1/168, 1/504, 1/1260, 1/2772, 1/5544, 1/12012, ...
(End)
T(n,k) is the number of edges of distance k from a fixed vertex in the n-dimensional hypercube. - Simon Burton, Nov 04 2022

			The triangle begins:
  1;
  1/2, 1/2;
  1/3, 1/6, 1/3;
  1/4, 1/12, 1/12, 1/4;
  1/5, 1/20, 1/30, 1/20, 1/5;
  ...
The triangle of denominators begins:
   1
   2   2
   3   6   3
   4  12  12    4
   5  20  30   20    5
   6  30  60   60   30    6
   7  42 105  140  105   42    7
   8  56 168  280  280  168   56    8
   9  72 252  504  630  504  252   72   9
  10  90 360  840 1260 1260  840  360  90  10
  11 110 495 1320 2310 2772 2310 1320 495 110 11

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, see 130.
B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 38.
G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See pp. 8-9.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993.
H. J. Brothers, Pascal's Prism: Supplementary Material.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Yilmaz Simsek, Construction of a generalization of the Leibnitz numbers and their properties, arXiv:2011.13701 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

Cf. A007622, A128064, A164555/A027642, A356546.
Cf. A094305, A121547, A121306, A119800, A002378, A007318.
Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816. A116071, A215652.
Denominators of i-th order differences of n^-1 are given in: (1st) A002378, (2nd) A027480, (3rd) A033488, (4th) A174002, (5th) A253946. - Louis Conover, Mar 02 2015
Columns k >= 1 (offset 1): A000027, A002378, A027480, A033488, A174002, A253946(n+4), ..., with sum of reciprocals: infinity, 1, 1/2, 1/3, 1/4, 1/5, ..., respectively. - Wolfdieter Lang, Jul 20 2022

a003506 n k = a003506_tabl !! (n-1) !! (n-1)
a003506_row n = a003506_tabl !! (n-1)
a003506_tabl = scanl1 (\xs ys ->
   zipWith (+) (zipWith (+) ([0] ++ xs) (xs ++ [0])) ys) a007318_tabl
a003506_list = concat a003506_tabl
-- Reinhard Zumkeller, Nov 14 2013, Nov 17 2011

with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008
A003506 := (n,k) -> k*binomial(n,k):
seq(print(seq(A003506(n,k),k=1..n)),n=1..7); # Peter Luschny, May 27 2011

L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Flatten[ Table[ 1 / L[n, m], {n, 1, 12}, {m, 1, n}]], 66]
t[n_, m_] = Gamma[n]/(Gamma[n - m]*Gamma[m]); Table[Table[t[n, m], {m, 1, n - 1}], {n, 2, 12}]; Flatten[%] (* Roger L. Bagula and Gary W. Adamson, Sep 14 2008 *)
Table[k*Binomial[n,k],{n,1,7},{k,1,n}] (* Peter Luschny, May 27 2011 *)
t[n_, k_] := Denominator[n!*k!/(n+k+1)!]; Table[t[n-k, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)

A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^(i-1)*(1-x)^(j-1)),x,1))

A(i,j)=if(i<1||j<1,0,1/sum(k=0,i-1,(-1)^k*binomial(i-1,k)/(j+k)))

{T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* Michael Somos, Feb 06 2011 */

T_row = lambda n: (n*(x+1)^(n-1)).list()
for n in (1..10): print(T_row(n)) # Peter Luschny, Feb 04 2017
# Assuming offset 0:
def A003506(n, k):
    return falling_factorial(n+1,n)//(factorial(k)*factorial(n-k))
for n in range(9): print([A003506(n, k) for k in range(n+1)]) # Peter Luschny, Aug 13 2022

a(n, 1) = 1/n; a(n, k) = a(n-1, k-1) - a(n, k-1) for k > 1.
Considering the integer values (rather than unit fractions): a(n, k) = k*C(n, k) = n*C(n-1, k-1) = a(n, k-1)*a(n-1, k-1)/(a(n, k-1) - a(n-1, k-1)) = a(n-1, k) + a(n-1, k-1)*k/(k-1) = (a(n-1, k) + a(n-1, k-1))*n/(n-1) = k*A007318(n, k) = n*A007318(n-1, k-1). Row sums of integers are n*2^(n-1) = A001787(n); row sums of the unit fractions are A003149(n-1)/A000142(n). - Henry Bottomley, Jul 22 2002
From Vladeta Jovovic, Nov 01 2003: (Start)
G.f.: x*y/(1-x-y*x)^2.
E.g.f.: x*y*exp(x+x*y). (End)
T(n,k) = n*binomial(n-1,k-1) = n*A007318(n-1,k-1). - Philippe Deléham, Aug 04 2006
Binomial transform of A128064(unsigned). - Gary W. Adamson, Aug 29 2007
From Roger L. Bagula and Gary W. Adamson, Sep 14 2008: (Start)
t(n,m) = Gamma(n)/(Gamma(n - m)*Gamma(m)).
f(s,n) = Integral_{x=0..oo} exp(-s*x)*x^n dx = Gamma(n)/s^n; t(n,m) = f(s,n)/(f(s,n-m)*f(s,m)) = Gamma(n)/(Gamma(n - m)*Gamma(m)); the powers of s cancel out. (End)
From Reinhard Zumkeller, Mar 05 2010: (Start)
T(n,5) = T(n,n-4) = A174002(n-4) for n > 4.
T(2*n,n) = T(2*n,n+1) = A005430(n). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(1,1) = 1 and, for n > 1, T(n,k) = 0 if k <= 1 or if k > n. - Philippe Deléham, Mar 17 2012
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,k+1-i). - Mircea Merca, Apr 11 2012
If we include a main diagonal of zeros so that the array is in the form
  0
  1   0
  2   2   0
  3   6   3   0
  4  12  12   4   0
...
then we obtain the exponential Riordan array [x*exp(x),x], which factors as [x,x]*[exp(x),x] = A132440*A007318. This array is the infinitesimal generator for A116071. A signed version of the array is the infinitesimal generator for A215652. - Peter Bala, Sep 14 2012
a(n,k) = (n-1)!/((n-k)!(k-1)!) if k > n/2 and a(n,k) = (n-1)!/((n-k-1)!k!) otherwise. [Forms 'core' for Pascal's recurrence; gives common term of RHS of T(n,k) = T(n-1,k-1) + T(n-1,k)]. - Jon Perry, Oct 08 2013
Assuming offset 0: T(n, k) = FallingFactorial(n + 1, n) / (k! * (n - k)!). The counterpart using the rising factorial is A356546. - Peter Luschny, Aug 13 2022

Edited by N. J. A. Sloane, Oct 07 2007

A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

Original entry on oeis.org

1, 2, 5, 8, 8, 13, 151, 256, 83, 146, 1433, 2588, 15341, 28211, 52235, 19456, 19345, 36362, 651745, 6168632, 1463914, 2786599, 122289917, 233836352, 140001721, 268709146, 774885169, 1491969394, 41711914513, 80530073893

Offset: 0

N. J. A. Sloane

The term a(12)=15341 is divisible by 23^2. Is there another term a(n) divisible by the square of a prime p larger than n+1? - M. F. Hasler, Jul 17 2012

			1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
G. Letac, Problèmes de probabilités, Presses Universitaires de France (1970), p. 14.
F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

T. D. Noe, Table of n, a(n) for n = 0..200
T. Mansour, Gamma function, beta function and combinatorial identities.
T. Sillke, More information
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Combinatorics, 14 (1993), 351-353.

Cf. A003149, A046826, A048211, A051389, A100512, A100513.

[Numerator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021

Numerator/@Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}]  (* Harvey P. Dale, Apr 21 2011 *)

P=1;vector(30,n,numerator(P)+0*P=P/2/n*(n+1)+1) \\ M. F. Hasler, Jul 17 2012

A046825(n)=numerator(sum(k=0,n,1/binomial(n,k))) \\ M. F. Hasler, Jul 19 2012

[numerator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021

Let P(n) = (1/n) * Sum_{k=0..n-1} 1/binomial(n-1, k) = A046878(n)/A046879(n) = A046825(n-1)/(n*A046826(n-1)): { 0, 1, 1, 5/6, 2/3, 8/15, ...}. Then P(n) = 2^(-n) * Sum_{k=1..n} 2^k / k = 2^(-n+1) * Sum_{k odd} binomial(n, k)/k; P(0) = 0, P(n) = P(n-1)/2 + 1/n. - Torsten Sillke (Torsten.Sillke(AT)uni-bielefeld.de)
G.f. for P(n): (2*log(1-z))/(-2+z). - Wouter Meeussen
P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
a(n) = numerator( A003149(n)/n! ). - G. C. Greubel, May 24 2021

References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de

A052186 Number of permutations of [n] with no strong fixed points.

Original entry on oeis.org

1, 0, 1, 3, 14, 77, 497, 3676, 30677, 285335, 2928846, 32903721, 401739797, 5298600772, 75092880273, 1138261010851, 18378421938366, 314928827507717, 5708689036074089, 109145365739197964, 2195167574579322013, 46331767712354136479, 1023970009016490622478

Offset: 0

N. J. A. Sloane, Feb 04 2000

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Equals INVERTi transform of the factorials, n starting with 0. Triangle A144108 has row sums = n! with left border = A052186. - Gary W. Adamson, Sep 11 2008

Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49

Alois P. Heinz, Table of n, a(n) for n = 0..450
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

Cf. A006932, A201684, A256168.
Cf. A144108, A000142. - Gary W. Adamson, Sep 11 2008
Column k=0 of A186373.

t1 := add(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 20 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
# second Maple program:
a:= proc(n) a(n):= -`if`(n<0, 1, add(a(n-i-1)*i!, i=0..n)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 21 2013

m = 20; CoefficientList[ Series[ 1 / (x + 1/Sum[ n!*x^n, {n, 0, m}]), {x, 0, m}], x] (* Jean-François Alcover, Aug 30 2011, after Michael Somos *)
nmax = 25; Rest[CoefficientList[Assuming[Element[x, Reals], Series[-1/(ExpIntegralEi[1/x]/E^(1/x) + 1), {x, 0, nmax+1}]], x]] (* Vaclav Kotesovec, Aug 05 2015 *)

{a(n)=if(n<0, 0, polcoeff( 1/ (x+1/sum(k=0, n, k!*x^k, x*O(x^n))), n))} /* Michael Somos, Oct 11 2006 */

G.f.: F(x)/(1 + x*F(x)), F(x) = Sum_{n >= 0} n!*x^n.
a(0)=1, a(1)=0, a(n) = (n-2)*a(n-1) + Sum_{k=0..n-1} a(k)*a(n-1-k) + Sum_{k=0..n-2} a(k)*a(n-2-k) if n > 1. - Michael Somos, Oct 11 2006
G.f.: 1/(1-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2/(1-... (continued fraction). - Paul Barry, Dec 09 2009
If p[i] = Stirling1(i,1) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i=j+1), and A[i,j]=0 otherwise, then, for n >= 1, a(n-1) = (-1)^(n-1) det A. - Milan Janjic, May 08 2010
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = upper left term in (-1)*M^(n+1), M = an infinite square production matrix in which a column of (-1)'s is prepended to Pascal's triangle as follows:
  -1, 1, 0, 0, 0, 0, ...
  -1, 1, 1, 0, 0, 0, ...
  -1, 1, 2, 1, 0, 0, ...
  -1, 1, 3, 3, 1, 0, ...
  -1, 1, 4, 6, 4, 1, ...
  ... (End)
G.f.: A(x) = 1/(1/G(0) + x); G(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 29 2011
G.f.: A(x) = 1/x = 1/(1+x)*(1+x/((1+x)*G(0)-x)); G(k) = 1 + x*(k+1) - x*(k+2)/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Dec 29 2011
G.f.: 1/(G(0) + x) where G(k) = 1  - x*(k+1)/(1 - x*(k+1)/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1/(1 - W(0)) where W(k) = x*(2*k+1) - 1 - x^2*(k+1)^2/W(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1/(G(0) + x), where G(k)= 1 + x*k - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
a(n) ~ n! * (1 - 2/n + 1/n^2 - 1/n^3 - 9/n^4 - 59/n^5 - 474/n^6 - 4560/n^7 - 50364/n^8 - 625385/n^9 - 8622658/n^10), for coefficients see A256168. - Vaclav Kotesovec, Mar 16 2015
a(n) = n! - Sum_{k=0..n-1} (n-k-1)!*a(k). - Pontus von Brömssen, Jul 10 2021
a(n) + A006932(n) = n!. - Pontus von Brömssen, Jul 10 2021

Better description from James Sellers, Mar 13 2000

A046662 Sum of mistyped version of binomial coefficients.

Original entry on oeis.org

1, 2, 7, 52, 749, 17686, 614227, 29354312, 1844279257, 147273109354, 14561325802271, 1745720380045852, 249461639720702917, 41886684733511640062, 8164388189339113521259, 1828191138807263097870256, 466057478369217965809683377, 134193343258948416556377786322

Offset: 0

Len Smiley

Origin of the name of this sequence: Binomial coefficients are n!/((n-k)!*k!) but if parentheses are omitted in the denominator, the formula might result in n!/(n-k)!*k! = n!*k!/(n-k)! and the sum giving a(n) instead of 2^n. If k! is forgotten altogether, one gets A000522. - Olivier Gérard, Mar 05 2024

Binomial transform of (n!)^2. - Peter Luschny, May 31 2014

Seiichi Manyama, Table of n, a(n) for n = 0..253
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

Cf. A003149, A064570, A229464.

Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)

Table[Sum[(n!k!)/(n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 29 2012 *)

a(n) = Sum_{k=0..n} n!*k!/(n-k)!.
E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - Ralf Stephan, Apr 02 2004
a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - Vladeta Jovovic, Jul 15 2004
From Peter Bala, Nov 26 2017: (Start)
a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
More generally, a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n). It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, ... with exact period 10. (End)
G.f.: Sum_{k>=0} (k!)^2*x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019
a(n) ~ (n!)^2. - Vaclav Kotesovec, May 03 2021
a(n) = 3F0(1,1,-n;;-1). - R. J. Mathar, Jun 26 2024

Corrected and extended by Harvey P. Dale, Sep 29 2012

A098361 Multiplication table of the factorial numbers read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 24, 120, 24, 12, 12, 24, 120, 720, 120, 48, 36, 48, 120, 720, 5040, 720, 240, 144, 144, 240, 720, 5040, 40320, 5040, 1440, 720, 576, 720, 1440, 5040, 40320, 362880, 40320, 10080, 4320, 2880, 2880, 4320, 10080, 40320, 362880

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - Stephen Crowley, Mar 28 2007
Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - Roger L. Bagula, Oct 29 2008
The triangle T(n, k) = k!*(n-k)! appears as denominators in the coefficients of the Niven polynomials x^n*(1 - x)^n/n! = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!). These polynomials are used in a proof that Pi^2 (hence Pi) is irrational. See the Niven and Havil references. - Wolfdieter Lang, May 07 2018; corrected by Dimitri Papadopoulos, Nov 30 2023
The case T(n+1,k) = k!*(n-k+1)!, 1 <= k <= n+1, n >= 0 is the number of choices for forming a cluster (compact group) of k numbered items arranged in a line on a set of permutations of n numbered items arranged in a line. - Igor Victorovich Statsenko, Oct 13 2023
The numbers T(n,k) also appear in the denominators of the partial fraction expansion of 1/(x*(x+1)*...*(x+n)) = Sum_{k=0..n} (-1)^k * 1/(T(n,k)*(x+k)). - Dimitri Papadopoulos, Nov 30 2023
It follows from the previous comment that the numbers T(n,k) also appear in the denominators of the coefficients of the logarithms of the integral of 1/(x*(x+1)*...*(x+n)): c + Sum{k=0...n} (-1)^k * 1/(T(n,k)) * ln(x+k). - Colin Linzer, Dec 18 2024

			The array A(n, k) starts in row n=0 with columns k >= 0 as:
       1,      1,      2,       6,      24,      120, ...
       1,      1,      2,       6,      24,      120, ...
       2,      2,      4,      12,      48,      240, ...
       6,      6,     12,      36,     144,      720, ...
      24,     24,     48,     144,     576,     2880, ...
     120,    120,    240,     720,    2880,    14400, ...
     720,    720,   1440,    4320,   17280,    86400, ...
    5040,   5040,  10080,   30240,  120960,   604800, ...
   40320,  40320,  80640,  241920,  967680,  4838400, ...
  362880, 362880, 725760, 2177280, 8709120, 43545600, ...
  ...
The triangle T(n, k) begins:
n\k       0      1     2     3     4     5     6     7     8      9      10...
0:        1
1:        1      1
2:        2      1     2
3:        6      2     2     6
4:       24      6     4     6    24
5:      120     24    12    12    24   120
6:      720    120    48    36    48   120   720
7:     5040    720   240   144   144   240   720  5040
8:    40320   5040  1440   720   576   720  1440  5040 40320
9:   362880  40320 10080  4320  2880  2880  4320 10080 40320 362880
10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800
... - _Wolfdieter Lang_, May 07 2018

R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

Stefano Spezia, First 101 antidiagonals of the array, flattened
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
I. V. Statsenko, Problem on variants of cluster formation at permutations in ordered structures, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2023, pp. 7-10. In Russian.

Row sums A003149.

Cf. A003991, A098358, A098359, A098360.

F:=Factorial; [F(n-k)*F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2022

seq(print(seq(k!*(n-k)!,k=0..n)),n=0..6); # Peter Luschny, Aug 23 2010

Table[(n+1)!*Beta[n-k+1, k+1], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Oct 29 2008 *)

f=factorial; flatten([[f(n-k)*f(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 12 2022

T(n, k) = k!*(n-k)! =  n!/C(n,k), (0<=k<=n). - Peter Luschny, Aug 23 2010
Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - R. J. Mathar, Dec 10 2010
E.g.f. as array: 1/((1 - x)*(1 - y)). - Stefano Spezia, Jul 10 2020

A006932 Number of permutations of [n] with at least one strong fixed point (a permutation p of {1,2,...,n} is said to have j as a strong fixed point if p(k) < j for k < j and p(k) > j for k > j).

Original entry on oeis.org

1, 1, 3, 10, 43, 223, 1364, 9643, 77545, 699954, 7013079, 77261803, 928420028, 12085410927, 169413357149, 2544367949634, 40758600588283, 693684669653911, 12499734669634036, 237734433597317987, 4759174459355303521

Offset: 1

N. J. A. Sloane

a(n) is also the number of permutation graphs with domination number one. See Definition 2.1, Lemma 2.3, and page 16 in the paper provided in the link by Theresa Baren, et al. - Daniel A. McGinnis, Oct 16 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.
K. Wayland, personal communication.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, and Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

Cf. A003149, A052186, A145878.

t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 1 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # James Sellers, Mar 13 2000

m = 22; s = Sum[n!*x^n, {n, 0, m}]; Range[0, m-1]! - CoefficientList[ Series[ s/(1+x*s), {x, 0, m}], x][[1;;m]] // Rest (* Jean-François Alcover, Apr 28 2011, after Maple code *)

a(n) ~ 2 * (n-1)! * (1 - 1/(2*n) + 1/(2*n^2) + 9/(2*n^3) + 59/(2*n^4) + 237/n^5 + 2280/n^6 + 25182/n^7 + 625385/(2*n^8) + 4311329/n^9 + 65375943/n^10). - Vaclav Kotesovec, Mar 17 2015
a(n) = Sum_{k=1..n} (n-k)!*A145878(k-1,0). See the link by Theresa Baren, et al. - Daniel A. McGinnis, Oct 15 2018
a(n) = A003149(n-1) - Sum_{k=0..n-1} (n-k-1)!*a(k). (This follows immediately from the preceding formula since A145878(k,0) = k! - a(k).) - Pontus von Brömssen, Jul 10 2021
a(n) + A052186(n) = n! - Pontus von Brömssen, Jul 10 2021

More terms from James Sellers, Mar 13 2000

Edited by Emeric Deutsch, Oct 29 2008

cp's OEIS Frontend

A174662 Partial sums of A003149.

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A182430 a(n) = n! - A003149(n-1).

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A084938 Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

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A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.

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A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

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A052186 Number of permutations of [n] with no strong fixed points.

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