A000629 -id:A000629 - cp's OEIS frontend

A108694 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, . . . ] DELTA [2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, . . . ] where DELTA is the operator defined in A084938.

1, 1, 2, 3, 9, 6, 13, 54, 69, 26, 75, 399, 747, 573, 150, 541, 3508, 8638, 9998, 5393, 1082, 4683, 35817, 109248, 169038, 139143, 57585, 9366, 47293, 416762, 1515531, 2935222, 3256907, 2064534, 691645, 94586

Offset: 0

Philippe Deléham, Jun 18 2005

Related to preferential arrangements of n elements (A000670) and necklaces of sets of labeled beads (A000629).

			1; 1, 2; 3, 9, 6; 13, 54, 69, 26; 75, 399, 747, 573, 150; ...

Cf. A000629, A000670, A032031, A084938.

Sum_{ k>=0 } T(n, k) = n!*3^n = A032031(n).

T(n, 0) = A000670(n); T(n, n) = A000629(n).

A136246 a(n) = (1/n!)Sum_{k=0..n} (-1)^(n-k)Stirling1(n,k)*A062208(k).

Original entry on oeis.org

1, 1, 32, 2712, 449102, 122886128, 50225389432, 28670796914144, 21789885975738524, 21271115441652577064, 25938193213744579451420, 38638907727108476424404864, 69044758685363149615280762608, 145768622491129079115419544343808, 358961215083489204505055286181798208

Offset: 0

Vladeta Jovovic, Mar 16 2008

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A330942.

Cf. A062208, A121316.

A000629 := proc(n) local k ; sum( k^n/2^k,k=0..infinity) ; end: A062208 := proc(n) option remember ; local a,stir,ni,n1,n2,n3,stir2,i,j,tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp),0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1,ni) ; n2 := op(2,ni) ; n3 := op(3,ni) ; a := a+combinat[multinomial](n,n1,n2,n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: A136246 := proc(n) local k ; add((-1)^(n-k)*combinat[stirling1](n,k)*A062208(k),k=0..n)/n! ; end: seq(A136246(n),n=0..14) ; # R. J. Mathar, Apr 01 2008

a[n_] := Sum[Binomial[Binomial[j, 3] + n - 1, n] * Sum[(-1)^(i - j)* Binomial[i, j], {i, j, 3n}], {j, 0, 3n}];
a /@ Range[0, 14] (* Jean-François Alcover, Feb 13 2020, after Andrew Howroyd *)

a(n) = {sum(j=0, 3*n, binomial(binomial(j,3)+n-1, n) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

a(n) = Sum_{m>=0} binomial(binomial(m,3)+n-1,n)/2^(m+1).

a(n) = Sum_{j=0..3*n} binomial(binomial(j,3)+n-1, n) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

More terms from R. J. Mathar, Apr 01 2008

Terms a(13) and beyond from Andrew Howroyd, Feb 09 2020

A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 7, 12, 6, 0, 1, 15, 50, 60, 24, 0, 1, 31, 180, 390, 360, 120, 0, 1, 63, 602, 2100, 3360, 2520, 720, 0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 0, 1, 511, 18660

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Row n gives the coefficients which express the sums of the n-th powers of the integers as a linear combination of binomial coefficients, thus:
     Sum_{k=1..r} k^n = A103438(n+r,r) = Sum_{k=0..n} T(n+1,k) * C(r,k),
  where, by convention, C(r,k) = 0 whenever r < k. - Robert B Fowler, Jan 16 2023

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    2;
  0, 1,   7,   12,     6;
  0, 1,  15,   50,    60,    24;
  0, 1,  31,  180,   390,   360,   120;
  0, 1,  63,  602,  2100,  3360,  2520,   720;
  0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040;
  ...

Peter Luschny, Row 0 to 44, recomputed Jan 19 2019

Column k = 0 is A000007.
Cf. A028246, A163626, A000629 (row sums).
Cf. A103438, A007318 (binomial coefficients).

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A142071row := proc(n) 1 - log(1 - x*(exp(z) - 1)):
series(%, z, 12): CL(n!*coeff(%, z, n), x) end:
for n from 0 by 1 to 7 do A142071row(n) od;
# Alternative:
A142071Row := proc(n) if n=0 then [1] else
CL(convert(series(polylog(-n+1, z/(1+z)), z, n*2), polynom), z) fi end:
seq(A142071Row(n), n=0..6); # Peter Luschny, Sep 06 2018

T[n_, k_] := If[k==0, Floor[1/(n + 1)], (k - 1)!*StirlingS2[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 06 2024 *)

Row n gives the coefficients of the polynomial defined by p(x, 0) = 1 and for n > 0 p(x, n) = Sum_{k >= 0} k^(n-1)*(x/(1 + x))^k = PolyLog(-n+1, x/(1+x)).

T(n, k) = (k - 1)! * Stirling2(n, k) for k > 0. - Detlef Meya, Jan 06 2024

Edited, T(0,0) = 1 prepended and new name by Peter Luschny, Sep 06 2018

A201338 E.g.f.: log((2 - exp(x))/(3 - 2*exp(x))).

Original entry on oeis.org

1, 4, 24, 196, 2040, 25924, 390264, 6804676, 135033720, 3007364164, 74315818104, 2018441506756, 59776933889400, 1917312391176004, 66216538949389944, 2449977966210378436, 96685769287005577080, 4053944607498740773444, 179973441341757042161784, 8433644996370680262923716

Offset: 1

Paul D. Hanna, Dec 03 2011

nonn

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 196*x^4/4! + 2040*x^5/5! +...
Note that A(x) = G(G(x)) where G(x) is an e.g.f. of A000629:
G(x) = x + 2*x^2/2! + 6*x^3/3! + 26*x^4/4! + 150*x^5/5! + 1082*x^6/6! +...

Cf. A000629, A201731.

Rest[CoefficientList[Series[Log[(2-E^x)/(3-2*E^x)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(log((2-exp(x+x*O(x^n)))/(3-2*exp(x+x*O(x^n)))),n)}

E.g.f.: G(G(x)) where G(x) = log(1/(2-exp(x))) is an e.g.f. of A000629 (with offset 1), where A000629(n) is the number of necklaces of partitions of n+1 labeled beads.
E.g.f.: log(1+x) o x/(1-2*x) o exp(x)-1, a composition of functions.
a(n) ~ (n-1)! * (1/log(3/2))^n. - Vaclav Kotesovec, May 23 2013

A201731 a(n) = [x^n/n!] log( (n - (n-1)exp(x)) / (n+1 - nexp(x)) ).

Original entry on oeis.org

1, 4, 54, 1544, 75750, 5676492, 603041334, 86210654224, 15958892198070, 3713676157320020, 1061084890984465446, 365202873520507832856, 149027843082185351506950, 71144948740332156241755868, 39282974873393643411310747350, 24840594864924259316810487005216

Offset: 1

Paul D. Hanna, Dec 04 2011

nonn

The function log((n - (n-1)*exp(x))/(n+1 - n*exp(x))) equals the n-th iteration of log(1/(2-exp(x))), the e.g.f. of A000629 (with offset 1), where A000629(n) is the number of necklaces of partitions of n+1 labeled beads.

			Let G(x) = log(1/(2-exp(x))) then the coefficients of x^n/n! in the k-th iteration of G(x) begin:
k=1: [(1), 2, 6, 26, 150, 1082, 9366, 94586, ..., A000629(n-1), ...];
k=2: [1,(4), 24, 196, 2040, 25924, 390264, 6804676, ..., A201338(n), ...];
k=3: [1, 6,(54), 654, 9990, 184686, 4015494, 100531374, ...];
k=4: [1, 8, 96, (1544), 31200, 760328, 21721056, 712459784, ...];
k=5: [1, 10, 150, 3010, (75750), 2295010, 81378150, 3307983010, ...];
k=6: [1, 12, 216, 5196, 156600, (5676492), 240593976, 11679764556, ...];
k=7: [1, 14, 294, 8246, 289590, 12224534, (603041334), 34053651926, ...];
k=8: [1, 16, 384, 12304, 493440, 23777296, 1338417024, (86210654224), ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence.

Cf. A000629, A201338, A201732.

{a(n)=n!*polcoeff(log((n - (n-1)*exp(x+x*O(x^n)))/(n+1 - n*exp(x+x*O(x^n)))),n)}

a(n) = (n+1) * A201732(n+1).

A202687 Triangle arising in the computation of hypersigma, definition 2 (A191161).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 18, 1, 1, 8, 36, 104, 1, 1, 10, 60, 260, 750, 1, 1, 12, 90, 520, 2250, 6492, 1, 1, 14, 126, 910, 5250, 22722, 65562, 1, 1, 16, 168, 1456, 10500, 60592, 262248, 756688, 1, 1, 18, 216, 2184, 18900, 136332, 786744, 3405096, 9825030, 1

Offset: 0

Alonso del Arte, Dec 22 2011

Given a squarefree number with n distinct prime factors put through the hypersigma function (A191161), each smaller divisor contributes a given number of "loose" 1s. Row n of this triangle tells how many loose 1s are contributed by prime divisors (column 1), by semiprime divisors (column 2), sphenic divisors (column 3) and so on up to column n - 1.

It may appear somewhat artificial that the leftmost and rightmost columns are all filled with 1s since under the hypersigma function, the smallest divisor of a number, 1, may be said to contribute two loose 1s (itself and the 1 in the recursion level immediately below). However, the reason for attributing one of these 1s to the number itself and the 1 in the recursion level below to that 1 in the recursion level above is to more clearly show how this triangle is tied to Pascal's triangle.

			Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1,  6, 18,   1;
  1,  8, 36, 104,    1;
  1, 10, 60, 260,  750,    1;
  1, 12, 90, 520, 2250, 6492, 1;

Jinyuan Wang, Rows n = 0..100 of triangle, flattened

Cf. A000629 (row sums).

A202687 := proc(n,k)
    if k = 0 or k = n then
        1;
    else
        binomial(n,k)*add(procname(k,j),j=0..k) ;
    end if;
end proc: # R. J. Mathar, Mar 15 2013

a[0, k_] := 1; a[n_, n_] := 1; a[n_, k_] := a[n, k] = Binomial[n, k] Sum[a[k, j], {j, 0, k}]; ColumnForm[Table[a[n, k], {n, 0, 9}, {k, 0, n}], Center]

T(n, k) = binomial(n, k)*Sum_{j=0..k} T(k, j).

A221961 Number of n X n symmetric lonesum ternary matrices.

Original entry on oeis.org

1, 3, 18, 149, 1390, 13377

Offset: 0

N. J. A. Sloane, Feb 05 2013

H. K. Kim, D. S. Krotov and J. Y. Lee, Matrices uniquely determined by their lonesums, Linear Algebra and its Applications, 438 (2013) no 7, 3107-3123

Cf. A000629 (the binary case).

See also A299906, A299907 for the general binary case.

Kim et al. (2013) give a formula.

A227044 a(n) = Sum_{k>=1} k^(2*n)/(2^k).

Original entry on oeis.org

1, 6, 150, 9366, 1091670, 204495126, 56183135190, 21282685940886, 10631309363962710, 6771069326513690646, 5355375592488768406230, 5149688839606380769088406, 5916558242148290945301297750, 8004451519688336984972255078166, 12595124129900132067036747870669270

Offset: 0

Vaclav Kotesovec, Jun 29 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Bisection of A000629.

Cf. A080163.

Table[Sum[k^(2*n)/(2^k), {k, 1, Infinity}], {n, 0, 20}]
a[n_] := PolyLog[-2 n, 1/2]; a[0] = 1; Array[a, 15, 0] (* Peter Luschny, Sep 06 2020 *)

{a(n) = sum(k=0, 2*n, (-2)^k * k! * stirling(2*n, k,2) )}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ (2n)!/(log(2))^(2*n+1).
a(n) = Sum_{k=0..2*n} (-2)^k * k! * Stirling2(2*n, k). - Paul D. Hanna, Apr 15 2018
a(n) = A000629(2*n). - Christian Krause, Nov 22 2022

A285868 Row sums of triangle A285867.

Original entry on oeis.org

1, 1, 4, 20, 126, 962, 8646, 89546, 1051350, 13811642, 200866326, 3205348346, 55704133590, 1047489675962, 21195507649686, 459258707587706, 10610386574074710, 260385846630175802, 6764666952807962646

Offset: 0

Wolfdieter Lang, May 04 2017

See A285867 for details.

Cf. A000629, A285867.

a(n) = Sum_{k=0..n} A285867(n, k).

a(n) = 1 for n = 0 and A000629(n) - n! for n >= 1.

A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563

Offset: 0

Ilya Gutkovskiy, Sep 26 2017

A(n,k) is the k-th binomial transform of A000670 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! +  (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
    1,     1,     1,     1,     1,      1,  ...
    1,     2,     3,     4,     5,      6,  ...
    3,     6,    11,    18,    27,     38,  ...
   13,    26,    51,    94,   161,    258,  ...
   75,   150,   299,   582,  1083,   1910,  ...
  541,  1082,  2163,  4294,  8345,  15666,  ...

G. C. Greubel, Antidiagonals n = 0..50, flattened
N. J. A. Sloane, Transforms

Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.
Rows n=0..2 give A000012, A000027, A102305.
Main diagonal gives A292916.

R:=PowerSeriesRing(Rationals(), 50);
T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
A292915:= func< n,k | T(k,n-k) >;
[A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; k^n +add(
       binomial(n, j)*A(j, k), j=0..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017

Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023

def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
def A292915(n,k): return T(k,n-k)
flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(k*x)/(2 - exp(x)).

A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - Seiichi Manyama, Dec 25 2023

cp's OEIS Frontend

A108694 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, . . . ] DELTA [2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, . . . ] where DELTA is the operator defined in A084938.

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A136246 a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A062208(k).

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A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

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A201338 E.g.f.: log((2 - exp(x))/(3 - 2*exp(x))).

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A201731 a(n) = [x^n/n!] log( (n - (n-1)*exp(x)) / (n+1 - n*exp(x)) ).

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A202687 Triangle arising in the computation of hypersigma, definition 2 (A191161).

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Maple

Mathematica

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A221961 Number of n X n symmetric lonesum ternary matrices.

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A227044 a(n) = Sum_{k>=1} k^(2*n)/(2^k).

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A136246 a(n) = (1/n!)Sum_{k=0..n} (-1)^(n-k)Stirling1(n,k)*A062208(k).

A201731 a(n) = [x^n/n!] log( (n - (n-1)exp(x)) / (n+1 - nexp(x)) ).