A001705 -id:A001705 - cp's OEIS frontend

A006675 Number of paths through an array.

0, 0, 2, 15, 104, 770, 6264, 56196, 554112, 5973264, 69998400, 886897440, 12089295360, 176484597120, 2748022986240, 45472329504000, 796983880089600, 14751208762214400, 287543058350284800

Offset: 0

N. J. A. Sloane

			x*(1-x)^-2 * (-log(1-x)) = x^2 + (5/2)*x^3 + (13/3)*x^4 + (77/12)*x^5 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
A. M. Khidr and B. S. El-Desouky, A symmetric sum involving the Stirling numbers of the first kind, European J. Combin., 5 (1984), 51-54.

Cf. A000254, A001705.

a[n_] := n*n!*(HarmonicNumber[n]-1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Nov 28 2011 *)

a(n) = n*n! * (H(n) - 1) where H(n) = Sum_{k=1..n} 1/k.
E.g.f. A(x) = x*(1-x)^-2 * (-log(1-x)).
a(n) = A001705(n) - A000254(n). - Peter Bala, Feb 12 2019

More terms from Joe Keane (jgk(AT)jgk.org)

A022819 a(n) = floor(1/(n-1) + 2/(n-2) + 3/(n-3) + ... + (n-1)/1).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 8, 11, 13, 16, 19, 22, 25, 28, 31, 34, 38, 41, 44, 48, 51, 55, 59, 62, 66, 70, 74, 78, 81, 85, 89, 93, 97, 101, 106, 110, 114, 118, 122, 126, 131, 135, 139, 144, 148, 152, 157, 161, 166, 170, 174, 179, 183, 188, 193, 197, 202, 206, 211, 216

Offset: 0

Clark Kimberling

nonn

a(n) = A214075(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

			a(2) = floor(1/1) = 1;
a(3) = floor(1/2 + 2/1) = floor(5/2) = 2;
a(4) = floor(1/3 + 2/2 + 3/1) = floor(26/6) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A027612.

import Data.Ratio ((%))
a022819 n = floor $ sum $ zipWith (%) [1 .. n-1] [n-1, n-2 .. 1]
-- Reinhard Zumkeller, Jul 03 2012

s=0; Table[s+=HarmonicNumber[j]//N; Floor[s],{j,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2010 *)
Join[{0},Floor[Accumulate[HarmonicNumber[Range[0,60]]]]] (* Harvey P. Dale, Sep 16 2019 *)

a(n) = floor(sum_{i=2..n} n/i) = floor(A000027(n)*(A001008(n)/A002805(n)-1)) = floor(A006675(n)/A000142(n)) = floor(A001705(n-1)/A000142(n-1)). - Henry Bottomley, May 05 2001

A143947 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the right-to-left minima is k (1 <= k <= n*(n+1)/2).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 1, 2, 1, 0, 0, 0, 6, 2, 3, 7, 2, 3, 1, 0, 0, 0, 0, 24, 6, 8, 14, 27, 10, 9, 14, 3, 4, 1, 0, 0, 0, 0, 0, 120, 24, 30, 46, 68, 142, 41, 53, 50, 73, 23, 17, 23, 4, 5, 1, 0, 0, 0, 0, 0, 0, 720, 120, 144, 204, 270, 436, 834, 260, 256, 351, 310, 463, 148, 145, 118, 148, 40

Offset: 1

Emeric Deutsch, Sep 22 2008

Row n contains n(n+1)/2 entries, first n-1 of which are 0. Sum of entries in row n = n! = A000142(n).
Sum of entries in column n = A143948(n).
T(n,n) = (n-1)!.
Sum_{k=n..n(n+1)/2} k*T(n,k) = A001705(n).

			T(4,6) = 3 because we have 4132, 3142 and 2143 with right-to-left minima at positions 2 and 4.
Triangle starts:
  1;
  0,  1,  1;
  0,  0,  2,  1,  2,  1;
  0,  0,  0,  6,  2,  3,  7,  2,  3,  1;
  0,  0,  0,  0, 24,  6,  8, 14, 27, 10,  9, 14,  3,  4,  1;
  ...

Alois P. Heinz, Rows n = 1..50, flattened

Cf. A000142, A000217, A001705, A143946, A143948.
T(n,2n) gives A368678.
Row maxima give A367594.

P:=proc(n) options operator, arrow: sort(expand(product(t^(n-j)+j,j=0..n-1))) end proc: for n to 7 do seq(coeff(P(n),t,i),i=1..(1/2)*n*(n+1)) end do; # yields sequence in triangular form

T[n_] := CoefficientList[Product[n-k+t^k, {k, 1, n-1}] t^(n-1), t];
Array[T, 10] // Flatten (* Jean-François Alcover, Feb 14 2021 *)

Generating polynomial of row n is (n-1+t)(n-2+t^2)(n-3+t^3)...(1+t^(n-1))t^n.

A067176 A triangle of generalized Stirling numbers: sum of consecutive terms in the harmonic sequence multiplied by the product of their denominators.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 11, 5, 1, 0, 50, 26, 7, 1, 0, 274, 154, 47, 9, 1, 0, 1764, 1044, 342, 74, 11, 1, 0, 13068, 8028, 2754, 638, 107, 13, 1, 0, 109584, 69264, 24552, 5944, 1066, 146, 15, 1, 0, 1026576, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 0, 10628640

Offset: 0

Henry Bottomley, Jan 09 2002

In the Coupon Collector's Problem with n types of coupon, the expected number of coupons required until there are only k types of coupon uncollected is a(n,k)*k!/(n-1)!.
If n+k is even, then a(n,k) is divisible by (n+k+1). For n>=k and k>= 0, a(n,k) = (n-k)!*H(k+1,n-k), where H(m,n) is a generalized harmonic number, i.e., H(0,n) = 1/n and H(m,n) = Sum_{j=1..n} H(m-1,j). - Leroy Quet, Dec 01 2006
This triangle is the same as triangle A165674, which is generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n), minus the first right hand column. - Johannes W. Meijer, Oct 16 2009

			Rows start 0; 1,0; 3,1,0; 11,5,1,0; 50,26,7,1,0; 274,154,47,9,1,0 etc. a(5,2) = 3*4*5*(1/3 + 1/4 + 1/5) = 4*5 + 3*5 + 3*4 = 20 + 15 + 12 = 47.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Columns are A000254, A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562, A051564, etc.

Cf. A093905 and A165674. - Johannes W. Meijer, Oct 16 2009

T[0, k_] := 1; T[n_, k_] := T[n, k] = Sum[ i*k^(i - 1)*Abs[StirlingS1[n - k, i]], {i, 1, n - k}]; Table[T[n,k], {n,1,10}, {k,1,n}] (* G. C. Greubel, Jan 21 2017 *)

a(n, k) = (n!/k!)*Sum_{j=k+1..n} 1/j = (A000254(n) - A000254(k)*A008279(n, n-k))/A000142(k) = a(n-1, k)*n + (n-1)!/k! = (a(n, k-1)-n!/k!)/k.

a(n, k) = Sum_{i=1..n-k} i*k^(i-1)*abs(stirling1(n-k, i)). - Vladeta Jovovic, Feb 02 2003

A093905 Triangle read by rows: for 0 <= k < n, a(n, k) is the sum of the products of all subsets of {n-k, n-k+1, ..., n} with k members.

Original entry on oeis.org

1, 1, 3, 1, 5, 11, 1, 7, 26, 50, 1, 9, 47, 154, 274, 1, 11, 74, 342, 1044, 1764, 1, 13, 107, 638, 2754, 8028, 13068, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 1, 19, 242, 2414, 19524, 127860

Offset: 1

Amarnath Murthy, Apr 24 2004

Triangle A165674, which is the reversal of this triangle, is generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). - Johannes W. Meijer, Oct 16 2009

			Triangle begins:
1
1 3
1 5 11
1 7 26 50
1 9 47 154 274
...
a(5, 3) = 4*3*2+5*3*2+5*4*2+5*4*3 = 154.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

The leading diagonal is given by A000254, Stirling numbers of first kind. The next nine diagonals are A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562 and A051564, generalized Stirling numbers.
Cf. A001705, A001711, A067176, A093344, A105954.
A165674 is the reversal of this triangle. - Johannes W. Meijer, Oct 16 2009

T[n_, 0] := 1; T[n_, k_]:= Product[i, {i, n - k, n}]*Sum[1/i, {i, n - k, n}]; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] (* G. C. Greubel, Jan 21 2017 *)

a(n, k) = prod(i=n-k, n, i)*sum(i=n-k,n,1/i);
tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jan 21 2017

a(n, k) = (Product_{i=n-k..n} i)*(Sum_{i=n-k..n} 1/i), where a(n, 0) = 1.

a(n, k) = A067176(n, n-k-1) = A105954(k+1, n-k). Row sums are given by A093344.

Edited and extended by David Wasserman, Apr 24 2007

A067948 Triangle of labeled rooted trees according to the number of increasing edges.

Original entry on oeis.org

1, 1, 1, 2, 5, 2, 6, 26, 26, 6, 24, 154, 269, 154, 24, 120, 1044, 2724, 2724, 1044, 120, 720, 8028, 28636, 42881, 28636, 8028, 720, 5040, 69264, 319024, 655248, 655248, 319024, 69264, 5040, 40320, 663696, 3793212, 10095228, 13861809, 10095228, 3793212, 663696, 40320

Offset: 1

Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 19 2002

Each line is symmetric.
The sum of each line is n^(n-1), A000169.
The outer diagonal is (n-1)!, A000142.
The next-to-last diagonal is A001705.

			Triangle starts:
   1;
   1,   1;
   2,   5,   2;
   6,  26,  26,   6;
  24, 154, 269, 154,  24;
  ...
From _Bruno Berselli_, Jan 12 2021: (Start)
The rows of the triangle are the coefficients of the following polynomials:
1: 1;
2: 1*x+1;
3: (x+2)*(2*x+1) = 2*x^2 + 5*x + 2;
4: (x+3)*(2*x+2)*(3*x+1) = 6*x^3 + 26*x^2 + 26*x + 6;
5: (x+4)*(2*x+3)*(3*x+2)*(4*x+1) = 24*x^4 + 154*x^3 + 269*x^2 + 154*x + 24, etc.
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.7.2), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Ira M. Gessel and Seunghyun Seo, A refinement of Cayley's formula for trees, Electronic J. Combin. 11, no. 2 (2004-6) (The Stanley Festschrift volume).
A. M. Khidr and B. S. El-Desouky, A symmetric sum involving the Stirling numbers of the first kind, European J. Combin., 5 (1984), 51-54.

Cf. A000142, A000169, A001705.

b:= proc(n) option remember;
      expand(x*mul(n-k+k*x, k=1..n-1))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 26 2024

L := CoefficientList[InverseSeries[Series[(Exp[-x y] + Sinh[x] - Cosh[x])/(1 - y), {x, 0, 8}]], {x}]; Table[CoefficientList[L, y][[n + 1]] n!, {n, 1, 8}] // Flatten (* Peter Luschny, Jun 23 2018 *)

G.f. of row n: Sum_{k=0..n-1} T(n, k) x^k = Product_{i=1..n-1} (n - i + i*x).
From Peter Bala, Sep 29 2011: (Start)
E.g.f.: Compositional inverse of (exp(x) - exp(x*t))/((1 - t)*exp(x*(1 + t))) = x + (1 + t)*x^2/2! + (2 + 5*t + 2*t^2)*x^3/3! + ...
Let f(x,t) = (1 - t)/(exp(-x) - t*exp(-x*t)) and let D be the operator f(x,t)*d/dx. Then the (n+1)-th row generating polynomial equals (D^n)(f(x,t)) evaluated at x = 0. See [Drake, example 1.7.2] for the combinatorial interpretation of this table in terms of labeled trees. (End)

A109792 Expansion of e.g.f. log(1+x)/(1-x)^2.

Original entry on oeis.org

1, 3, 14, 70, 444, 3108, 25584, 230256, 2342880, 25771680, 312888960, 4067556480, 57424792320, 861371884800, 13869128448000, 235775183616000, 4264876094976000, 81032645804544000, 1627055289796608000, 34168161085728768000, 754132445894209536000, 17345046255566819328000

Offset: 1

Vladeta Jovovic, Aug 14 2005

G. C. Greubel, Table of n, a(n) for n = 1..400

Cf. A001705, A092692.

CoefficientList[Series[Log[1+x]/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
a[n_] := n! ((-1)^n (n + 1) LerchPhi[-1, 1, n + 2] + Log[2] (n + 1) + ((-1)^(n + 1) - 1) /2); Table[Simplify[a[n]], {n, 1, 22}] (* Peter Luschny, Jun 22 2022 *)

for(n=1,25, print1(n!*sum(k=1,n, sum(i=1, k, (-1)^(i+1)/i)), ", ")) \\ G. C. Greubel, Jan 21 2017

a(n) = n!*Sum_{k=1..n} Sum_{i=1..k} (-1)^(i+1)/i.
a(n) ~ n!*n*log(2). - Vaclav Kotesovec, Jun 27 2013
a(n) = n!*((-1)^n*(n + 1)*LerchPhi(-1, 1, n + 2) + log(2)*(n + 1) + ((-1)^(n + 1) - 1) / 2). - Peter Luschny, Jun 22 2022

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b-1)/2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

A163937 Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)).

Original entry on oeis.org

1, 1, 2, 2, 10, 3, 6, 52, 43, 4, 24, 308, 472, 136, 5, 120, 2088, 4980, 2832, 369, 6, 720, 16056, 53988, 49808, 13638, 918, 7, 5040, 138528, 616212, 826160, 381370, 57540, 2167, 8, 40320, 1327392, 7472952, 13570336, 9351260, 2469300, 222908, 4948, 9

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher-order exponential integral E(x,m=2,n) lead to triangle A028421, see A163931 for information on E(x,m,n). The o.g.f.s. of the right-hand columns of triangle A028421 have a nice structure: gf(p) = W2(z,p)/(1-z)^(2*p) with p = 1 for the first right-hand column, p = 2 for the second right-hand column, etc. The coefficients of the W2(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001147 (minus a(0)), see A163936 for more information.

			The first few W2(z,p) polynomials are
W2(z,p=1) = 1/(1-z)^2;
W2(z,p=2) = (1 +  2*z)/(1-z)^4;
W2(z,p=3) = (2 + 10*z +  3*z^2)/(1-z)^6;
W2(z,p=4) = (6 + 52*z + 43*z^2 + 4*z^3)/(1-z)^8.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums equal A001147 (n>=1).
A000142, 2*A001705, are the first two left hand columns.
A000027 is the first right hand column.
Cf. A163931 (E(x,m,n)) and A028421.
Cf. A163936 (E(x,m=1,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*((m - k)/1!)*Binomial[2*n, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0,m-1, (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k) *stirling1(m+n-k-1,m-k)), ", "))) \\ G. C. Greubel, Aug 13 2017

a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*Stirling1(m+n-k-1,m-k), 1 <= m <= n.

A334670 a(n) = (2n+1)!! (Sum_{k=1..n} 1/(2*k+1)).

Original entry on oeis.org

0, 1, 8, 71, 744, 9129, 129072, 2071215, 37237680, 741975345, 16236211320, 387182170935, 9995788416600, 277792140828825, 8269430130712800, 262542617405726175, 8855805158351474400, 316285840413064454625, 11924219190760084593000, 473245342972281190686375, 19722890048636406588957000

Offset: 0

Seiichi Manyama, Sep 10 2020

nonn

			a(1) = 3 * (1/3) = 1.
a(2) = 3*5 * (1/3 + 1/5) = 8.
a(3) = 3*5*7 * (1/3 +1/5 + 1/7) = 71.

Seiichi Manyama, Table of n, a(n) for n = 0..403

Column k=1 of A335095.

Cf. A001147, A001705, A004041, A288875, A334000, A334066.

a[n_] := (2*n + 1)!! * Sum[1/(2*k + 1), {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 29 2021 *)

{a(n) = prod(k=1, n, 2*k+1)*sum(k=1, n, 1/(2*k+1))}

{a(n) = if(n<2, n, 4*n*a(n-1)-(2*n-1)^2*a(n-2))}

a(n) + A001147(n+1) = A004041(n).
a(n) = (2*n+1) * a(n-1) + A001147(n) for n>0.
P-finite with recurrence a(n) = 4*n*a(n-1) - (2*n-1)^2 * a(n-2) for n>1.

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A006675 Number of paths through an array.

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A022819 a(n) = floor(1/(n-1) + 2/(n-2) + 3/(n-3) + ... + (n-1)/1).

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A143947 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the right-to-left minima is k (1 <= k <= n*(n+1)/2).

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A067176 A triangle of generalized Stirling numbers: sum of consecutive terms in the harmonic sequence multiplied by the product of their denominators.

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A093905 Triangle read by rows: for 0 <= k < n, a(n, k) is the sum of the products of all subsets of {n-k, n-k+1, ..., n} with k members.

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A067948 Triangle of labeled rooted trees according to the number of increasing edges.

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A109792 Expansion of e.g.f. log(1+x)/(1-x)^2.

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A334670 a(n) = (2n+1)!! (Sum_{k=1..n} 1/(2*k+1)).