A341587 -id:A341587 - cp's OEIS frontend

A039814 Matrix square of Stirling-1 triangle A008275.

1, -2, 1, 7, -6, 1, -35, 40, -12, 1, 228, -315, 130, -20, 1, -1834, 2908, -1485, 320, -30, 1, 17582, -30989, 18508, -5005, 665, -42, 1, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Exponential Riordan array [1/((1 + x)*(1 + log(1 + x))), log(1 + log(1 + x))]. The row sums of the unsigned array give A007840 (apart from the initial term). - Peter Bala, Jul 22 2014

Also the Bell transform of (-1)^n*A003713(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
      1;
     -2,    1;
      7,   -6,     1;
    -35,   40,   -12,   1;
    228, -315,   130, -20,   1;
  -1834, 2908, -1485, 320, -30, 1;
...

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)

Column k=1..3 give (-1)^(n-1) * A003713(n), (-1)^n * A341587(n), (-1)^(n-1) * A341588(n).
Cf. A007840.
Cf. A039810, A039815, A039816, A039817.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*add(k!*abs(Stirling1(n+1,k+1)), k=0..n), 10); # Peter Luschny, Jan 28 2016

max = 9; t = Table[StirlingS1[n, k], {n, 1, max}, {k, 1, max}]; t2 = t.t; Table[t2[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)
rows = 9;
t = Table[(-1)^n*Sum[k!*Abs[StirlingS1[n+1, k+1]], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Feb 13 2022

E.g.f. of k-th column: ((log(1+log(1+x)))^k)/k!.
E.g.f.: 1/(1 + t)*( 1 + log(1 + t) )^(x-1) = 1 + (-2 + x)*t + (7 - 6*x + x^2)*t^2/2! + .... - Peter Bala, Jul 22 2014
T(n,k) = Sum_{j=0..n} Stirling1(n,j) * Stirling1(j,k). - Seiichi Manyama, Feb 13 2022

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

Original entry on oeis.org

1, 12, 130, 1485, 18508, 253400, 3805723, 62437500, 1113510409, 21479997957, 446094038806, 9930796412082, 236037249893092, 5968192832899412, 160007282538148508, 4534905316824903144, 135500246340709682692, 4257646241716404353684, 140366073694357927723936, 4845119946789226304526392

Offset: 3

Ilya Gutkovskiy, Feb 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..418

Cf. A000399, A000559, A003713, A008275, A039814, A341587.

nmax = 22; CoefficientList[Series[-Log[1 + Log[1 - x]]^3/6, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 3]], {k, 3, n}], {n, 3, 22}]

a(n) = Sum_{k=3..n} |Stirling1(n, k) * Stirling1(k, 3)|.

a(n) ~ (n-1)! * log(n)^2 / (2 * (1 - exp(-1))^n) * (1 + (2*gamma - 2*log(exp(1) - 1)) / log(n) + (gamma^2 - Pi^2/6 - 2*log(exp(1) - 1)*gamma + log(exp(1)-1)^2) / log(n)^2). - Vaclav Kotesovec, Jun 04 2022

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 0

Peter Luschny, Jun 27 2019

			Triangle starts:
[0] [1]
[1] [0,       1]
[2] [0,      -2,        1]
[3] [0,       7,       -6,       1]
[4] [0,     -35,       40,     -12,        1]
[5] [0,     228,     -315,     130,      -20,      1]
[6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
[7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
[8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
[9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 8.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).

Cf. A039814 (variant), A129062, A325873.

p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

def a_row(n):
    s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

From Seiichi Manyama, Apr 18 2025: (Start)
T(n,k) = Sum_{j=k..n} Stirling1(n,j) * Stirling1(j,k).
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + log(1 + x)). (End)

A341575 E.g.f.: log(1 - log(1 - x))^2 / 2.

Original entry on oeis.org

1, 0, 4, 5, 58, 217, 2035, 13470, 134164, 1243770, 14129410, 164244808, 2151576620, 29671566836, 444758323628, 7055358559376, 119546765395744, 2139179551573104, 40486788832168944, 805969129348431936, 16860672502118423136, 369459637224850523808, 8467140450141232328160

Offset: 2

Ilya Gutkovskiy, Feb 15 2021

nonn

Cf. A000254, A000558, A008275, A079642, A081048, A089064, A302547, A302548, A341587.

nmax = 24; CoefficientList[Series[Log[1 - Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[Abs[StirlingS1[n, k]] StirlingS1[k, 2], {k, 2, n}], {n, 2, 24}]

a(n) = Sum_{k=2..n} |Stirling1(n, k)| * Stirling1(k, 2).

a(n) = (-1)^n * Sum_{k=2..n} Stirling1(n, k) * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.

A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.

Original entry on oeis.org

1, 2, 40, 1485, 81088, 5856900, 526685269, 56704848200, 7112345477952, 1018548226480356, 163987811350464660, 29321558852248050388, 5764958268855541178967, 1236150756215397667568170, 287086392921014590422630300, 71789589754855255636302048525, 19231403740347427723119910379040

Offset: 0

Ilya Gutkovskiy, Feb 15 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A003713, A008275, A039814, A321712, A341587, A341588.

Table[Sum[Abs[StirlingS1[2 n, k] StirlingS1[k, n]], {k, n, 2 n}], {n, 0, 16}]
Table[((2 n)!/n!) SeriesCoefficient[(-Log[1 + Log[1 - x]])^n, {x, 0, 2 n}], {n, 0, 16}]

a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1)*stirling(k, n, 1))); \\ Michel Marcus, Feb 16 2021

a(n) = ((2*n)!/n!) * [x^(2*n)] (-log(1 + log(1 - x)))^n.
From Vaclav Kotesovec, Feb 15 2021: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = -16*p*q^2 * log(-2*q/(1+r))^(1+r) / ((1 + 2*q + r)^2 * (1 + 1/(p*(1+r)))^r) = 17.84101281316291323354184111891200669611476053165484517795417711039479218...
p = LambertW(-1, -1/(exp(1/(1+r))*(1+r)))
q = LambertW(-1, -(1+r)/exp((1+r)/2)/2)
r = 0.5094050884976689299791685259225203723646676600942448390861428232759777841...
is the root of the equation (1+p)*(1+r)^2 * (1 + 2*q + r) * log(-p*(1+r)) + 2*log(-(1+r)/(2*q)) * ((1+q)*(1 + p + p*r) - (1+r) * log(-p*(1+r)) * (p - q + r + p*r + (1+p) * (1+q) * (1+r) * (log(1 + 1/(p*(1+r))) - log(-log(-(1+r)/(2*q)))))) = 0
and c = 0.1417076025518808268972093339771762801784527709... (End)

A351525 Expansion of e.g.f. (log(1 + log(1 + log(1+ x))))^2 / 2.

Original entry on oeis.org

1, -9, 87, -975, 12657, -188090, 3159699, -59326371, 1232843529, -28116615263, 698649506871, -18796044698977, 544507930693022, -16903759793180115, 559960766050363931, -19719027513960290370, 735696883534117583082, -28991986984973263419262

Offset: 2

Seiichi Manyama, Feb 13 2022

sign

Column 2 of A039815.

Cf. A000268, A341587, A351513, A351526, A351527.

With[{nn=20},CoefficientList[Series[Log[1+Log[1+Log[1+x]]]^2/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 15 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(log(1+log(1+log(1+x)))^2/2))

T(n, k) = if(k==0, n==1, sum(j=0, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = (-1)^n*sum(k=1, n-1, binomial(n-1, k)*T(k, 3)*T(n-k, 3));

a(n) = (-1)^n * Sum_{k=1..n-1} binomial(n-1,k) * A000268(k) * A000268(n-k).

A351526 Expansion of e.g.f. (log(1 + log(1 + log(1 + log(1+ x)))))^2 / 2.

Original entry on oeis.org

1, -12, 152, -2210, 36976, -704837, 15132932, -362099010, 9566898126, -276863733707, 8715530417502, -296641340905299, 10858928017129838, -425542158316462627, 17779220784851800828, -789053832262002586555, 37076561046965367191298

Offset: 2

Seiichi Manyama, Feb 13 2022

sign

Column 2 of A039816.

Cf. A000310, A341587, A351514, A351525, A351527.

my(N=20, x='x+O('x^N)); Vec(serlaplace(log(1+log(1+log(1+log(1+x))))^2/2))

T(n, k) = if(k==0, n==1, sum(j=0, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = (-1)^n*sum(k=1, n-1, binomial(n-1, k)*T(k, 4)*T(n-k, 4));

a(n) = (-1)^n * Sum_{k=1..n-1} binomial(n-1,k) * A000310(k) * A000310(n-k).

A351527 Expansion of e.g.f. (log(1 + log(1 + log(1 + log(1 + log(1+ x))))))^2 / 2.

Original entry on oeis.org

1, -15, 235, -4200, 86020, -2001055, 52305780, -1520815230, 48747603100, -1709228504170, 65115320810260, -2679459929923699, 118482699493123571, -5604477255138004835, 282449438671531808676, -15111729578894643263239

Offset: 2

Seiichi Manyama, Feb 13 2022

sign

Column 2 of A039817.

Cf. A000359, A341587, A351515, A351525, A351526.

my(N=20, x='x+O('x^N)); Vec(serlaplace(log(1+log(1+log(1+log(1+log(1+x)))))^2/2))

T(n, k) = if(k==0, n==1, sum(j=0, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = (-1)^n*sum(k=1, n-1, binomial(n-1, k)*T(k, 5)*T(n-k, 5));

a(n) = (-1)^n * Sum_{k=1..n-1} binomial(n-1,k) * A000359(k) * A000359(n-k).

A383171 Expansion of e.g.f. log(1 + log(1 - 2*x)/2)^2 / 2.

Original entry on oeis.org

0, 0, 1, 9, 91, 1090, 15298, 247352, 4537132, 93195696, 2120623984, 52973194560, 1441635171040, 42464913775232, 1346297567292416, 45715740985471744, 1655552663185480448, 63698261991541393408, 2595107348458704209920, 111613055867327344582656

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Cf. A383170, A383172.

Cf. A341587, A383163.

a(n) = sum(k=2, n, 2^(n-k)*abs(stirling(n, k, 1)*stirling(k, 2, 1)));

a(n) = Sum_{k=2..n} 2^(n-k) * |Stirling1(n,k) * Stirling1(k,2)|.

a(n) ~ sqrt(Pi) * log(n) * 2^(n + 1/2) * n^(n - 1/2) / (exp(1) - exp(-1))^n * (1 + (gamma + log(2) - log(exp(2)-1))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Apr 18 2025

cp's OEIS Frontend

A039814 Matrix square of Stirling-1 triangle A008275.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A341575 E.g.f.: log(1 - log(1 - x))^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A341589 a(n) = Sum_{k=n..2*n} |Stirling1(2*n, k) * Stirling1(k, n)|.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351525 Expansion of e.g.f. (log(1 + log(1 + log(1+ x))))^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A351526 Expansion of e.g.f. (log(1 + log(1 + log(1 + log(1+ x)))))^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A351527 Expansion of e.g.f. (log(1 + log(1 + log(1 + log(1 + log(1+ x))))))^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.