A001705 -id:A001705 - cp's OEIS frontend

A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.

1, 9, 130, 2900, 93576, 4141872, 241353792, 17929776384, 1655071418880, 185914776960000, 24978180045312000, 3955930130221056000, 729464836964806656000, 154952762244805582848000, 37566943754471090749440000, 10310706109241121091092480000

Offset: 1

Leroy Quet, May 07 2001

Sum of generalized harmonic numbers squared multiplied by (n!)^2. agenh(n) = Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(n, j) = Sum_{k = 1..n} 1/k^j. - Alexander Adamchuk, Oct 27 2004

			a(3) = 6^2 *(1 + (1 + 1/2^2) + (1 + 1/2^2 + 1/3^2)) = 130.

Harry J. Smith, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001705, A007406, A007407.

[(Factorial(n))^2*(&+[(1+j)/(n-j)^2: j in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Apr 09 2021

A060944:= n-> (n!)^2*add((1+j)/(n-j)^2, j=0..n-1); seq(A060944(n), n=1..15); # G. C. Greubel, Apr 09 2021

Table[(n!)^2*Sum[(k+1)/(n-k)^2, {k, 0, n-1}], {n, 1, 10}]

a(n)={n!^2 * sum(k=1, n, sum(j=1, k, 1/j^2))} \\ Harry J. Smith, Jul 15 2009

[(factorial(n))^2*sum((1+j)/(n-j)^2 for j in (0..n-1)) for n in (1..15)] # G. C. Greubel, Apr 09 2021

From Alexander Adamchuk, Oct 27 2004: (Start)
a(n) = (n!)^2 * Sum_{k=0..n-1} (k+1)/(n-k)^2.
a(n) = (n!)^2 * Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(k, 2) = A007406(k) / A007407(k). (End)
Sum_{n>=1} a(n) * x^n / (n!)^2 = polylog(2,x) / (1 - x)^2. - Ilya Gutkovskiy, Jul 15 2020

A081525 Erroneous version of A027612.

Original entry on oeis.org

1, 5, 13, 77, 87, 223, 481, 4609, 4861, 55991, 58301

Offset: 1

dead

Cf. A001705.

a(n) = A001705(n) / gcd(A001705(n), n!). Note: A001705 starts with n=0. - Martin Fuller, Jan 03 2006

Corrected and extended by Martin Fuller, Jan 03 2006

A081526 Erroneous version of A027611.

Original entry on oeis.org

1, 2, 3, 12, 10, 20, 35, 280, 252, 2520, 2310

Offset: 1

dead

Cf. A001705.

n! / gcd(A001705(n), n!). Note: A001705 starts with n=0 - Martin Fuller, Jan 03 2006

Corrected and extended by Martin Fuller, Jan 03 2006

A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.

Original entry on oeis.org

0, 2, 14, 94, 684, 5508, 49104, 482256, 5185440, 60668640, 767940480, 10462227840, 152698210560, 2377651449600, 39350097561600, 689874448435200, 12773427499929600, 249097496204390400, 5103595024496640000, 109608397522606080000, 2462475687669043200000

Offset: 1

Roger L. Bagula, Sep 14 2006

nonn

Former title (corrected): A Legendre-based recurrence sequence. Let b(n) = ((4*n+2)*x -(2*n+1) )/(n+1)*b(n-1) - (n/(n+1))*b(n-2), where x=1, then a(n) = (n+1)!*b(n)/6. - G. C. Greubel, Oct 03 2019

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 782

G. C. Greubel, Table of n, a(n) for n = 1..445
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].

Cf. A001008, A002805, A001705, A001711.

List([1..30], n-> Factorial(n+1)*(Sum([1..n+1], k-> 1/k) - 3/2) ); # G. C. Greubel, Oct 03 2019

[Factorial(n+1)*(HarmonicNumber(n+1) - 3/2): n in [1..30]]; // G. C. Greubel, Oct 03 2019

a:=n-> (n+1)!*add(1/k,k=3..n+1): seq(a(n),n=1..30); # Gary Detlefs, Jul 15 2010

x=1; b[1]:=0; b[2]:=2; b[n_]:= b[n]= ((-2*n-1) +(4*n+2)*x)/(n+1)*b[n-1] - (n/(n+1))*b[n-2]; Table[b[n]*(n+1)!/6, {n,30}]
Table[(n+1)!*(HarmonicNumber[n+1] - 3/2), {n,30}] (* G. C. Greubel, Oct 03 2019 *)

vector(30, n, (n+1)!*(sum(k=1,n+1, 1/k) - 3/2) ) \\ G. C. Greubel, Oct 03 2019

[factorial(n+1)*(harmonic_number(n+1) - 3/2) for n in (1..30)] # G. C. Greubel, Oct 03 2019

Let b(n) = ((-2*n-1) +(4*n+2)*x)/(n+1)*b(n-1) - (n/(n+1))*b(n-2) with x=1, then a(n) = b(n)*(n+1)!/6.
a(n) = (n+1)! * Sum_{k=3..n+1} 1/k. - Gary Detlefs, Jul 15 2010
a(n) = 2*A001711(n-2) for n >= 2. - Pontus von Brömssen, Jan 04 2025

If all terms are really negative, sequence should probably be negated. - N. J. A. Sloane, Oct 01 2006

Negated terms and edited by G. C. Greubel, Oct 03 2019

A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 2, 3, 2, 4, 3, 1, 2, 5, 3, 3, 2, 2, 1, 3, 3, 3, 1, 1, 2, 2, 2, 5, 2, 2, 2, 5, 1, 3, 4, 4, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2, 6, 2, 3, 4, 2, 4, 2, 3, 3, 2, 4, 4, 4, 3, 3, 3, 3, 4, 2, 3, 4, 2, 2, 5, 3, 2, 2, 4, 4, 2, 2, 1, 6, 4, 2, 5, 3, 5, 1, 2, 2, 3, 4, 2, 3, 3, 3, 5

Offset: 1

Jonathan Vos Post, Nov 09 2006

We must include multiplicity in the definition due to terms such as a(16) = 29889983 = 19 * 31^2 * 1637. The primes are those n for which a(n) = Omega(A027612(n))= 1, namely a(2) = 5, a(3) = 13, a(6) = 223, a(9) = 4861, a(18) = 197698279, a(25) = 25472027467. The semiprimes are those for which a(n) = 2, such as when n = 4, 5, 7, 8, 11, 12, 13, 15, 19, 23, 24. The 3-almost primes are those for which a(n) = 3, as with the "3-brilliant" a(10) = 55991 = 13 * 59 * 73, a(14), a(17), a(21), a(22), a(26).

			a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.

G. C. Greubel, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Harmonic Number, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.

Cf. A001222 Number of prime divisors of n (counted with multiplicity), A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, A027611, A001008, A002805, A001705, A006675, A093418.

PrimeOmega[Numerator[Table[Sum[k/(n - k + 1), {k, 1, n}], {n, 1, 50}]]] (* G. C. Greubel, Jan 22 2017 *)

a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).

A140713 Triangle read by rows: T(n,k) is the number of white corners of rank k in all the permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

Original entry on oeis.org

1, 5, 1, 26, 9, 2, 154, 70, 26, 6, 1044, 562, 268, 102, 24, 8028, 4860, 2700, 1308, 504, 120, 69264, 45756, 28224, 15828, 7728, 3000, 720

Offset: 2

Emeric Deutsch, May 29 2008

Sum of row n is A140712(n).
T(n,0) = A001705(n-1).
T(n,n-2) = (n-2)!.

			Triangle starts:
1;
5,1;
26,9,2;
154,70,26,6;
1044,562,268,102,24;

K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.

K. Eriksson and S. Linusson, The size of Fulton's essential set, Electronic J. Combinatorics, Vol. 2, #R6, 1995.
K. Eriksson and S. Linusson, Combinatorics of Fulton's essential set, ResearchGate, 1998.

Cf. A001705, A140711, A140712.

A376582 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 5, 1, 26, 7, 1, 154, 47, 9, 1, 1044, 342, 74, 11, 1, 8028, 2754, 638, 107, 13, 1, 69264, 24552, 5944, 1066, 146, 15, 1, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 6999840, 2592720, 662640, 127860, 19524, 2414, 242, 19, 1, 80627040, 30334320, 7893840, 1557660, 245004, 31594, 3382, 299, 21, 1

Offset: 0

Igor Victorovich Statsenko, Sep 29 2024

			Triangle starts:
[0]       1;
[1]       5,       1;
[2]      26,       7,       1;
[3]     154,      47,       9,        1;
[4]    1044,     342,      74,       11,       1;
[5]    8028,    2754,     638,      107,      13,     1;
[6]   69264,   24552,    5944,     1066,     146,    15,    1;
[7]  663696,  241128,   60216,    11274,    1650,   191,   17,    1;

Column k: A001705 (k=0), A001711 (k=1), A001716 (k=2), A001721 (k=3), A051524 (k=4), A051545 (k=5), A051560 (k=6).

Cf. A094587 and A173333 for m=0.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k): m:=1: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m,n,k) = Sum_{i=0..n-k} Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!, for m=1.

A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 11, 6, 0, 1, 7, 26, 50, 24, 0, 1, 9, 47, 154, 274, 120, 0, 1, 11, 74, 342, 1044, 1764, 720, 0, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 0, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 0, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880

Offset: 0

Ilya Gutkovskiy, Sep 21 2017

			E.g.f. of column k: A_k(x) = x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 6*k + 2)*x^3/3! + 2*(2*k^3 + 9*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
   0,    0,     0,     0,     0,      0,  ...
   1,    1,     1,     1,     1,      1,  ...
   1,    3,     5,     7,     9,     11,  ...
   2,   11,    26,    47,    74,    107,  ...
   6,   50,   154,   342,   638,   1066,  ...
  24,  274,  1044,  2754,  5944,  11274,  ...

Columns k=0..11 give A104150, A000254, A001705, A001711 (with offset 1), A001716 (with offset 1),  A001721 (with offset 1), A051524, A051545, A051560, A051562, A051564, A203147.
Rows n=0..3 give  A000004, A000012, A005408, A080663 (with offset 0).
Main diagonal gives A058806.

Table[Function[k, n! SeriesCoefficient[-Log[1 - x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: -log(1 - x)/(1 - x)^k.

A300910 Expansion of e.g.f. 1/(1 - x)^(x/(1 - x)^2).

Original entry on oeis.org

1, 0, 2, 15, 116, 1070, 11754, 149436, 2145296, 34193736, 598061160, 11377384920, 233732130312, 5153974126704, 121354505626704, 3037419444974040, 80497938647953920, 2251124265581428800, 66225476356207660224, 2044005966844402035456, 66025689709572751040640, 2227221130525199246067840, 78301158190416233445985920

Offset: 0

Ilya Gutkovskiy, Mar 15 2018

nonn

Exponential transform of A006675.

			1/(1 - x)^(x/(1 - x)^2) = 1 + 2*x^2/2! + 15*x^3/3! + 116*x^4/4! + 1070*x^5/5! + 11754*x^6/6! + 149436*x^7/7! + ...

N. J. A. Sloane, Transforms

Cf. A000254, A001705, A006675, A027611, A027612, A087761, A300491.

a:=series(1/(1-x)^(x/(1-x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[1/(1 - x)^(x/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k k! (HarmonicNumber[k] - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: A(x) = exp(B(x)*C(x)), where B(x) is the g.f. of the sequence {0, 1, 2, 3, 4, 5, ...} and C(x) is the g.f. of the sequence {0, 1, 1/2, 1/3, 1/4, 1/5, ...}.

a(0) = 1; a(n) = Sum_{k=1..n} k*k!*(H(k)-1)*binomial(n-1,k-1)*a(n-k), where H(k) is the k-th harmonic number.

A336746 Triangle read by rows: T(n,k) = (n-k-1+H(k+1))*((k+1)!) for 0 <= k <= n where H(k+1) = Sum_{i=0..k} 1/(i+1) for k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 3, 5, 11, 26, 4, 7, 17, 50, 154, 5, 9, 23, 74, 274, 1044, 6, 11, 29, 98, 394, 1764, 8028, 7, 13, 35, 122, 514, 2484, 13068, 69264, 8, 15, 41, 146, 634, 3204, 18108, 109584, 663696, 9, 17, 47, 170, 754, 3924, 23148, 149904, 1026576, 6999840

Offset: 0

Werner Schulte, Aug 02 2020

			The triangle starts:
n\k :  0   1   2    3    4     5      6       7        8        9
=================================================================
  0 :  0
  1 :  1   1
  2 :  2   3   5
  3 :  3   5  11   26
  4 :  4   7  17   50  154
  5 :  5   9  23   74  274  1044
  6 :  6  11  29   98  394  1764   8028
  7 :  7  13  35  122  514  2484  13068   69264
  8 :  8  15  41  146  634  3204  18108  109584   663696
  9 :  9  17  47  170  754  3924  23148  149904  1026576  6999840
...

Cf. A001477 (column 0), A005408 (column 1), A016969 (column 2), A001705 (main diagonal), A000254 (1st subdiagonal), A000774 (2nd subdiagonal).

T(n,k) = T(n,k-1) + k * T(n-1,k-1) for 0 < k <= n with initial values T(n,0) = n for n >= 0 and T(i,j) = 0 if j < 0 or j > i.
T(n,k) = k! + T(n-1,k-1) * (k+1) for 0 < k <= n.
T(n,k) = (k+1)! + T(n-1,k) for 0 <= k < n.
E.g.f. of main diagonal (case n=0) and n-th subdiagonal (n>0): Sum_{k>=0} T(n+k,k) * x^k / k! = (n - log(1-x)) / (1-x)^2 for n >= 0.
G.f. of column k>=0: Sum_{n>=k} T(n,k) * y^n = (T(k,k) * y^k + ((k+1)! - T(k,k)) * y^(k+1)) / (1-y)^2.
G.f.: Sum_{n>=0, k=0..n} T(n,k)*x^k*y^n/k! = (y - (1-y) * log(1-x*y)) / ((1-y)^2 * (1-x*y)^2).

cp's OEIS Frontend

A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.

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A081525 Erroneous version of A027612.

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A081526 Erroneous version of A027611.

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A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.

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A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

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A140713 Triangle read by rows: T(n,k) is the number of white corners of rank k in all the permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

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A376582 Triangle of generalized Stirling numbers.

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A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.

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