A165675 -id:A165675 - cp's OEIS frontend

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 2, 6, 24, 1, 1, 1, 2, 6, 24, 120, 1, 1, 1, 2, 6, 24, 120, 720, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880

Offset: 1

Johannes W. Meijer, Oct 05 2009

			Triangle starts:
1,
1, 1,
1, 1, 1,
1, 1, 1, 2,
1, 1, 1, 2, 6,
1, 1, 1, 2, 6, 24,
1, 1, 1, 2, 6, 24, 120,
1, 1, 1, 2, 6, 24, 120, 720,
1, 1, 1, 2, 6, 24, 120, 720, 5040,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...

Cf. A138771, A165675 and A000142.
A000012 (3x), A007395, A010722, A010863 equal the first six left hand columns.
A159333 equals, for n=>-1, all right hand columns.
A067078 equals the row sums.

nmax:=11: for n from 1 to nmax do a(n,1):=1 od: for n from 2 to nmax do for m from 2 to n do a(n,m):=(m-2)! od: od: for n from 1 to nmax do seq(a(n,m),m=1..n) od;

a(n) = A138771(n)/A165675(n-1).

A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Oct 05 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2+6*n+3*n^2)/x^2 - (6+22*n+18*n^2+ 4*n^3)/x^3 + ... ) is discussed in A028421. The formula for the asymptotic expansion leads for n = 1, 2, 3, .., to the left hand columns of the triangle given above.
The recurrence relations of the right hand columns of this triangle lead to Pascal's triangle A007318, their a(n) formulas lead to Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; cf. A080663, A165676, A165677, A165678 and A165679.
The row sums of this triangle lead to A093344. Surprisingly the e.g.f. of the row sums Egf(x) = (exp(1)*Ei(1,1-x) - exp(1)*Ei(1,1))/(1-x) leads to the exponential integrals in view of the fact that E(x,m=1,n=1) = Ei(n=1,x). We point out that exp(1)*Ei(1,1) = A073003.
The Maple programs generate the coefficients of the triangle given above. The first one makes use of a relation between the triangle coefficients, see the formulas, and the second one makes use of the asymptotic expansions of the E(x,m=2,n).
Amarnath Murthy discovered triangle A093905 which is the reversal of our triangle.
A165675 is an extended version of this triangle. Its reversal is A105954.
Triangle A094587 is generated by the asymptotic expansions of E(x,m=1,n).

A093905 is the reversal of this triangle.
A000254, A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562, A051564 are the first ten left hand columns.
A080663, n>=2, is the third right hand column.
A165676, A165677, A165678 and A165679 are the next right hand columns, A093344 gives the row sums.
A073003 is Gompertz's constant.
A094587 is generated by the asymptotic expansions of E(x, m=1, n).
Cf. A165675, A105954 (Quet) and A067176 (Bottomley).
Cf. A007318 (Pascal), A028421 (Wiggen), A126671 (Wood).

nmax:=9; for n from 1 to nmax do a(n, n) := 1 od: for n from 2 to nmax do a(n, 1) := n*a(n-1, 1) + (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m = 1..n), n = 1..nmax);
# End program 1
nmax := nmax+1: m:=2; with(combinat): EA := proc(x, m, n) local E, i; E:=0: for i from m-1 to nmax+2 do E := E + sum((-1)^(m+k1+1) * binomial(k1, m-1) * n^(k1-m+1) * stirling1(i, k1), k1=m-1..i) / x^(i-m+1) od: E:= exp(-x)/x^(m) * E: return(E); end: for n1 from 1 to nmax do f(n1-1) := simplify(exp(x) * x^(nmax+3) * EA(x, m, n1)); for m1 from 0 to nmax+2 do b(n1-1, m1) := coeff(f(n1-1), x, nmax+2-m1) od: od: for n1 from 0 to nmax-1 do for m1 from 0 to n1-m+1 do a(n1-m+2, m1+1) := abs(b(m1, n1-m1)) od: od: seq(seq(a(n, m), m = 1..n),n = 1..nmax-1);
# End program 2
# Maple programs revised by Johannes W. Meijer, Sep 22 2012

a(n,m) = (n-m+1)*a(n-1,m) + a(n-1,m-1), for 2 <= m <= n-1, with a(n,n) = 1 and a(n,1) = n*a(n-1,1) + (n-1)!.

a(n,m) = product(i, i= m..n)*sum(1/i, i = m..n).

A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

Original entry on oeis.org

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

Offset: 0

Leroy Quet, Jun 26 2005

Antidiagonal sums are A093345 (n! * (1 + Sum_{i=1..n}((1/i)*Sum_{j=0..i-1} 1/j!))). - Gerald McGarvey, Aug 27 2005
A recasting of A093905 and A067176. - R. J. Mathar, Mar 01 2009
The triangular array of this sequence is the reversal of A165675 which is related to the asymptotic expansion of the higher order exponential integral E(x,m=2,n); see also A165674. - Johannes W. Meijer, Oct 16 2009

			A(2, 2) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.
Array A(n, k) begins:
  [n\k]  0       1       2        3        4        5          6
  -------------------------------------------------------------------
  [0]    1,      1,      1,       1,       1,       1,         1, ...
  [1]    1,      3,      5,       7,       9,       11,       13, ...
  [2]    2,     11,     26,      47,      74,      107,      146, ...
  [3]    6,     50,    154,     342,     638,     1066,     1650, ...
  [4]   24,    274,   1044,    2754,    5944,    11274,    19524, ...
  [5]  120,   1764,   8028,   24552,   60216,   127860,   245004, ...
  [6]  720,  13068,  69264,  241128,  662640,  1557660,  3272688, ...
  [7] 5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, ...

G. C. Greubel, Table of n, a(n) for the first 27 rows, flattened
Arthur T. Benjamin, David Gaebler and Robert Gaebler, A Combinatorial Approach to Hyperharmonic Numbers, INTEGERS, Electronic Journal of Combinatorial Number Theory, Volum 3, #A15, 2003.

Column 0 = A000142 (factorial numbers).
Column 1 = A000254 (Stirling numbers of first kind s(n, 2)) starting at n=1.
Column 2 = A001705 (Generalized Stirling numbers: a(n) = n!*Sum_{k=0..n-1}(k+1)/(n-k)), starting at n=1.
Column 3 = A001711 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1)).
Column 4 = A001716 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1)).
Column 5 = A001721 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+1, 1)*5^k*stirling1(n+1, k+1)).
Column 6 = A051524 (2nd unsigned column of A051338) starting at n=1.
Column 7 = A051545 (2nd unsigned column of A051339) starting at n=1.
Column 8 = A051560 (2nd unsigned column of A051379) starting at n=1.
Column 9 = A051562 (2nd unsigned column of A051380) starting at n=1.
Column 10= A051564 (2nd unsigned column of A051523) starting at n=1.
2nd row is A005408 (2n - 1, starting at n=1).
3rd row is A080663 (3n^2 - 1, starting at n=1).
Main diagonal gives A384024.
Cf. A000254, A165674 and A165675, A028421 and A126671.

H := proc(n, k) option remember; if n = 0 then 1/k else add(H(n - 1, j), j = 1..k) fi end: A := (n, k) -> (n + 1)!*H(k, n + 1):
# Alternative with standard harmonic number:
A := (n, k) -> if k = 0 then n! else (harmonic(n + k) - harmonic(k - 1))*(n + k)! / (k - 1)! fi:
for n from 0 to 7 do seq(A(n, k), k = 0..6) od;
# Alternative with hypergeometric formula:
A := (n, k) -> (n+1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k+1], 1):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n=0..7); # Peter Luschny, Jul 01 2022

H[0, m_] := 1/m; H[n_, m_] := Sum[H[n - 1, k], {k, m}]; a[n_, m_] := m!H[n, m]; Flatten[ Table[ a[i, n - i], {n, 10}, {i, n - 1, 0, -1}]]
Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm (* to view the table *)
(* Robert G. Wilson v, Jun 27 2005 *)

a(n, k) = polcoef(prod(j=0, n, 1+(j+k)*x), n); \\ Seiichi Manyama, May 19 2025

A(n, k) = (Harmonic(n + k) - Harmonic(k - 1))*(n + k)!/(k - 1)! if k > 0, otherwise n!.
From Gerald McGarvey, Aug 27 2005, edited by Peter Luschny, Jul 02 2022: (Start)
E.g.f. for column k: -log(1 - x)/(x*(1 - x)^k).
Row 3 is r(n) = 4*n^3 + 18*n^2 + 22*n + 6.
Row 4 is r(n) = 5*n^4 + 40*n^3 + 105*n^2 + 100*n + 24.
Row 5 is r(n) = 6*n^5 + 75*n^4 + 340*n^3 + 675*n^2 + 548*n + 120.
Row 6 is r(n) = 7*n^6 + 126*n^5 + 875*n^4 + 2940*n^3 + 4872*n^2 + 3528*n + 720.
Row 7 is r(n) = 8*n^7 + 196*n^6 + 1932*n^5 + 9800*n^4 + 27076*n^3 + 39396*n^2 + 26136*n + 5040.
The sum of the polynomial coefficients for the n-th row is |S1(n, 2)|, which are the unsigned Stirling1 numbers which appear in column 1.
A(m, n) = Sum_{k=1..m} n*A094645(m, n)*(n+1)^(k-1). (A094645 is Generalized Stirling number triangle of first kind, e.g.f.: (1-y)^(1-x).) (End)
In Gerard McGarvey's formulas for the row coefficients we find Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; see A165674. - Johannes W. Meijer, Oct 16 2009
A(n, k) = (n + 1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k + 1], 1). - Peter Luschny, Jul 01 2022
A(n,k) = [x^n] Product_{j=0..n} (1 + (j+k)*x). - Seiichi Manyama, May 19 2025

More terms from Robert G. Wilson v, Jun 27 2005

Edited by Peter Luschny, Jul 02 2022

A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 5, 2, 24, 50, 26, 14, 6, 120, 274, 154, 94, 54, 24, 720, 1764, 1044, 684, 444, 264, 120, 5040, 13068, 8028, 5508, 3828, 2568, 1560, 720, 40320, 109584, 69264, 49104, 35664, 25584, 17520, 10800, 5040

Offset: 1

Emeric Deutsch, Apr 10 2008

T(n,0)=(n-1)!=A000142(n-1).
T(n,1)=A000254(n-1).
T(n,2)=A001705(n-2).
T(n,3)=2*A001711(n-4).
T(n,4)=6*A001716(n-5).
T(n,n-1)=(n-2)! (n>=2).
Sum(kT(n,k),k=0..n-1)=(n-1)!(n-1)(n+2)/4=A138772(n).

			T(4,2)=5 because we have (1)(23)(4), (1)(24)(3), (13)(24), (12)(34) and (14)(23).
Triangle starts;
1;
1,1;
2,3,1;
6,11,5,2;
24,50,26,14,6;
120,274,154,94,54,24;

Cf. A000142, A000254, A001705, A001711, A001716, A138772.
From Johannes W. Meijer, Oct 16 2009: (Start)
A000142 equals for n=>1 the row sums.
a(n) = A165680(n) * A165675(n-1).
(End)

T:=proc (n,k) if k = 0 then factorial(n-1) elif n <= k then 0 else (n-1)*T(n-1, k)+factorial(n-2) end if end proc: for n to 9 do seq(T(n, k), k=0..n-1) end do;

T(n,k)=(n-1)T(n-1,k)+(n-2)! (1<=k<=n-1). The row generating polynomials P[n](t) satisfy: P[n+1](t)=nP[n](t)+(n-1)!(t+t^2+...+t^n).

A203147 (n-1)-st elementary symmetric function of {11, 12, 13, 14, ..., 10 + n}.

Original entry on oeis.org

1, 23, 431, 7750, 140274, 2604744, 50046408, 998853264, 20742534576, 448372820160, 10086271796160, 235977273544320, 5737221760152960, 144817445022243840, 3791415072241843200, 102851265545431603200, 2888120485072388966400

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A001705, A165675.

t[n_] := Table[k + 10, {k, 1, n}];
a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}]

A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (kn+j)x).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 26, 6, 1, 7, 74, 342, 24, 1, 9, 146, 1650, 5944, 120, 1, 11, 242, 4578, 48504, 127860, 720, 1, 13, 362, 9774, 189144, 1763100, 3272688, 5040, 1, 15, 506, 17886, 520024, 9660840, 76223664, 97053936, 40320, 1, 17, 674, 29562, 1164024, 34201080, 586813968, 3817038960, 3270729600, 362880

Offset: 0

Seiichi Manyama, May 18 2025

			Square array begins:
    1,      1,       1,       1,        1,        1, ...
    1,      3,       5,       7,        9,       11, ...
    2,     26,      74,     146,      242,      362, ...
    6,    342,    1650,    4578,     9774,    17886, ...
   24,   5944,   48504,  189144,   520024,  1164024, ...
  120, 127860, 1763100, 9660840, 34201080, 93638820, ...

Columns k=0..3 give A000142, A384024, A383678, A382349.
Main diagonal gives A380707.
Cf. A165675.

a(n, k) = sum(j=0, n, (j+1)*(k*n)^j*abs(stirling(n+1, j+1, 1)));

A(n,k) = A165675((k+1)*n,k*n).
A(n,k) = Sum_{j=0..n} (j+1) * (k*n)^j * |Stirling1(n+1,j+1)|.
A(n,k) = (n+1)! * Sum_{j=0..n} (-1)^j * binomial(-k*n,j)/(n+1-j).
A(n,k) = ((k+1)*n)!/(k*n)! * (1 + k*n * Sum_{j=1..n} 1/(k*n+j)).

A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).

Original entry on oeis.org

1, 3, 26, 342, 5944, 127860, 3272688, 97053936, 3270729600, 123418922400, 5154170774400, 235977273544320, 11752173128586240, 632474276804697600, 36576553723886131200, 2261980049125982976000, 148956705206745595084800, 10406288081667512679321600, 768701832940487804295168000

Offset: 0

Vaclav Kotesovec, May 17 2025

nonn

Central terms of triangle A165675.

Cf. A000407, A201546, A383869.

Table[SeriesCoefficient[Product[1 + (n+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, (k+1)*n^k*abs(stirling(n+1, k+1, 1))); \\ Seiichi Manyama, May 18 2025

a(n) ~ n! * log(2) * 4^n * sqrt(n/Pi).
a(n) ~ log(2) * 2^(2*n + 1/2) * n^(n+1) / exp(n).
From Seiichi Manyama, May 18 2025: (Start)
a(n) = A165675(2*n,n).
a(n) = Sum_{k=0..n} (k+1) * n^k * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-n,k)/(n+1-k).
a(n) = (2*n)!/n! * (1 + n * Sum_{k=1..n} 1/(n+k)). (End)

A093345 a(n) = n! * {1 + Sum[i=1..n, 1/i*Sum(j=0..i-1, 1/j!)]}.

Original entry on oeis.org

1, 2, 6, 23, 108, 605, 3956, 29649, 250892, 2367629, 24662700, 281153801, 3482350724, 46572620757, 668943488084, 10271127486065, 167892667249116, 2911049382788189, 53365747562592092, 1031352659792534169

Offset: 0

Ralf Stephan, Apr 26 2004

nonn

Number of {12,2*1}-avoiding signed permutations in the hyperoctahedral group B_n.

D. Daly and L. Pudwell, Pattern avoidance in rook monoids, 2013. - From _N. J. A. Sloane_, Feb 03 2013
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.

Cf. A000774.
Cf. A000522, A093344.
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
Equals row sums of A165675.
(End)

a[n_] := n! (1+Sum[1/i Sum[1/j!, {j, 0, i-1}], {i, 1, n}])
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Oct 05 2018 *)

a(n)=n!+n!*sum(i=1,n,1/i*sum(j=0,i-1,1/j!))

E.g.f.: (exp(1)*(Ei(1, 1-x)-Ei(1, 1))+1)/(1-x). a(n) = n!*(1+Sum(A000522(i-1)/i!, i =1..n)). - Vladeta Jovovic, Apr 27 2004

Conjecture: a(n) -2*n*a(n-1) +(n^2-2)*a(n-2) -(n-2)^2*a(n-3)=0. - R. J. Mathar, May 30 2014

A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).

Original entry on oeis.org

3, -50, 1764, -109584, 10628640, -1486442880, 283465647360, -70734282393600, 22376988058521600, -8752948036761600000, 4148476779335454720000, -2342787216398718566400000, 1554454559147562279567360000

Offset: 1

Clark Kimberling, Dec 30 2011

sign

			The first 10 terms of the "full sequence" are as follows:
1, 3i, -11, -50i, 274, 1764i, -13068, -109584i, 1026576, 10628640i;
Abbreviate "elementary symmetric function" as esf. Then, starting with {i, 2i, 3i, 4i, ...}:
0th esf of {i}: 1
1st esf of {i, 2i}: i+2i = 3i
2nd esf of {i, 2i, 3i}: -2-3-6 = -11.
For the alternating terms 3i, -50i, ..., see A203240.

Cf. A000254, A094310, A165675, A203240.

f[k_] := k*I; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]
Table[-I*a[2 n], {n, 1, 22}]     (* A203239 *)
Table[a[2 n - 1], {n, 1, 22}]    (* A203240 *)
Table[(-1)^(n + 1)*(2*n)!*HarmonicNumber[2*n], {n, 13}] (* Arkadiusz Wesolowski, Mar 25 2013 *)

a(n) = (-1)^(n+1)*(2*n)!*Sum_{i=1..2n} 1/i. - Arkadiusz Wesolowski, Mar 25 2013
From Anton Zakharov, Oct 26 2016: (Start)
a(n) = (-1)^(n+1)*Sum_{k=1..n} A094310(2n,k).
(-1)^(n+1)*a(n) = A000254(2n) (signed bisection of A000254). (End)

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3n+k)x).

Original entry on oeis.org

1, 7, 146, 4578, 189144, 9660840, 586813968, 41283943344, 3299858098560, 295294500123840, 29242449106502400, 3174506423754019200, 374845813851886709760, 47828682507084551654400, 6557612642418946942310400, 961431335221085133398784000, 150095351600371197275428454400

Offset: 0

Seiichi Manyama, May 18 2025

nonn

Column k=3 of A382347.

Cf. A165675.

Table[SeriesCoefficient[Product[(1 + (3*n+k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2025 *)

a(n) = sum(k=0, n, (k+1)*(3*n)^k*abs(stirling(n+1, k+1, 1)));

a(n) = A165675(4*n,3*n).
a(n) = Sum_{k=0..n} (k+1) * (3*n)^k * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-3*n,k)/(n+1-k).
a(n) = (4*n)!/(3*n)! * (1 + 3*n * Sum_{k=1..n} 1/(3*n+k)).
a(n) ~ log(4/3) * 2^(8*n+1) * n^(n+1) / (exp(n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, May 23 2025

cp's OEIS Frontend

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

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A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).

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A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

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A203147 (n-1)-st elementary symmetric function of {11, 12, 13, 14, ..., 10 + n}.

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A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (k*n+j)*x).

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A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).

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A093345 a(n) = n! * {1 + Sum[i=1..n, 1/i*Sum(j=0..i-1, 1/j!)]}.

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A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).

A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (kn+j)x).

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3n+k)x).