A293037 -id:A293037 - cp's OEIS frontend

A334192 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (kj + 1)^n / ((-k)^j j!).

1, 1, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

			Square array begins:
   1,   1,    1,    1,    1,    1,  ...
   0,   0,    0,    0,    0,    0,  ...
  -1,  -2,   -3,   -4,   -5,   -6,  ...
  -1,  -4,   -9,  -16,  -25,  -36,  ...
   2,   4,    0,  -16,  -50, -108,  ...
   9,  64,  189,  384,  625,  864,  ...

Columns k=1..3 give A293037, A334190, A334191.

Cf. A309386, A334165, A334193 (diagonal).

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (1 - Exp[k x])/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (-x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).

E.g.f. of column k: exp(x + (1 - exp(k*x)) / k).

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k(nk + 1)^n/k!.

Original entry on oeis.org

1, 0, -3, 19, 497, -1899, -489491, -15433676, 618450881, 120846851155, 7012261819901, -467816186167659, -175527285590430863, -20961845760818684812, 568194037748383908653, 898095630359015975379151, 220433074470274983356464897, 16144974747716546214909454181

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

sign

Cf. A000587, A284860, A285065, A293037, A298373, A307066, A308645.

Table[Exp[1] Sum[(-1)^k (n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Table[n! SeriesCoefficient[Exp[1 + x - Exp[n x]], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] n^k BellB[k, -1], {k, 0, n}], {n, 1, 17}]]

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

a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000587(k).

A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0, 0, 1, 246, 1918, 1260, 0, 0, 0, 0, 0, 0, 1, 501, 6825, 9450, 945, 0, 0, 0, 0, 0, 0, 1, 1012, 22935, 56980, 17325, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 25 2022

{{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 0;
[2] 0, 1,   0;
[3] 0, 1,   0,    0;
[4] 0, 1,   3,    0,    0;
[5] 0, 1,  10,    0,    0,  0;
[6] 0, 1,  25,   15,    0,  0,  0;
[7] 0, 1,  56,  105,    0,  0,  0,  0;
[8] 0, 1, 119,  490,  105,  0,  0,  0,  0;
[9] 0, 1, 246, 1918, 1260,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

A008299 is an irregular subtriangle with more information.
A358622 (second order Stirling cycle numbers).
Cf. A000296 (row sums), alternating row sums (apart from sign): A000587, A293037, and A014182.
Cf. A048993, A201637, A264428, A269939, A340264.

T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j)*(-1)^(k - j),
j = 0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := exp(z*(exp(t) - t - 1)): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k = 0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358623 := proc(n, k) if n = 0 then return 1 fi;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, n - k - j - 1), j = 0..n-k-1)
end: seq(seq(A358623(n, k), k = 0..n), n = 0..11);

# recursion over rows
from functools import cache
@cache
def StirlingSetOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingSetOrd2(n - 2)
    row: list[int] = StirlingSetOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * rov[k - 1] + k * row[k]
    return row
for n in range(9): print(StirlingSetOrd2(n))
# Alternative, using function BellMatrix from A264428.
def f(k: int) -> int:
    return 1 if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(j, k - j)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = n!*[z^k][t^n] exp(z*(exp(t) - t - 1)).
T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(n, k - j)*{n - k + j, j}, where {n, k} denotes the Stirling set numbers.
T(n, k) = (n - 1) * T(n-2, k-1) + k * T(n-1, k) with suitable boundary conditions.
T(n + k, k) = A269939(n, k), which might be called the Ward set numbers.

A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 15, -6, 1, 52, -60, 30, -8, 1, -203, 260, -150, 50, -10, 1, 877, -1218, 780, -300, 75, -12, 1, -4140, 6139, -4263, 1820, -525, 105, -14, 1, 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1, -115975, 190323, -149040, 73668, -25578, 6552, -1260, 180, -18, 1

Offset: 0

Peter Luschny, Jun 18 2023

The triangle algorithm, as understood here, is a transformation that maps a sequence of integers (a(n) : n >= 0) to a polynomial sequence. A polynomial sequence is a sequence of polynomials (P(n,x) : n >= 0) with degree(P(n, x)) = n for all n >= 0.
The polynomials P(n, x) are recursively defined by P(n, x) = p(n, 0, x), where the initial sequence is p(0, m, x) = a(m), and for n > 0 is given by
  p(n, m, x) = (m + 1)*p(n - 1, m + 1, x) - (m + 1 - x)*p(n - 1, m, x).
Here we identify the polynomial sequence with the infinite lower triangular array of its coefficients, T(n, k) = [x^k] P(n, x). We call the mapping (a(n) : n >= 0) -> (T(n, k) : 0 <= k <= n) the 'triangle algorithm', following the lead of Kawasaki and Ohno.
Evaluating P(n, x) at different values of x gives rise to a multitude of other sequences; in particular, the transformation a(n) -> b(n) = P(n, 1) will be called the Akiyama-Tanigawa transform of a.
The triangle algorithm was studied by Akiyama and Tanigawa, Chen, Imatomi, Arakawa and Kaneko, Kawasaki and Ohno, and others, at first in connection with the Bernoulli and Poly-Bernoulli numbers.
.
The paradigmatic examples are:
  a(n) = 1 -> x^n, the standard base of polynomials, A023531.
  a(n) = n + 1 -> binomial(n, k), Pascal triangle, A007318.
  a(n) = n + 1 -> P(n, 1) powers of 2, A000079.
  a(n) = n + 1 -> P(n, 0) the all 1's sequence A000012.
  a(n) = 2^n -> [x^k] P(n, x), A154921.
  a(n) = 2^n -> P(n, 0) Fubini numbers, A000670.
  a(n) = 2^n -> P(n, 1) ordered set partitions of subsets of [n], A000629.
  a(n) = 2^n -> P(n,-1) osp. of [n] with even number of blocks, A052841.
  a(n) = 1 / (n + 1) -> [x^k] B(n, x), Bernoulli polynomials, A196838/A196839.
  a(n) = 1 / (n + 1) -> B(n, 1), the Bernoulli numbers, A164555/A027642.
  a(n) = Chen(n) -> skp(n, x), Swiss-Knife polynomials, A153641.
  a(n) = Chen(n) -> P(n, 0), 2^n*Euler(n, 1/2) = Euler(n), A122045.
  a(n) = Chen(n) -> P(n, 1), 2^n*Euler(n, 1), A155585.
  a(n) = (-1)^n/n! -> [x^k] P(n, x) this "Bell" triangle.
  a(n) = (-1)^n/n! -> (-1)^n*P(n, 1) = Bell(n), A000110.
  a(n) = (-1)^n/n! -> (-1)^n*P(n,-1) = 2-Bell(n), A005493.
  a(n) = 1/n! -> (-1)^n*P(n, 1) = complementary Bell(n), A000587.
  a(n) = 1/n! -> (-1)^n*P(n,-1) = complementary 2-Bell(n), A074051.
  (For Chen's sequence see A363524.)
.
The present sequence deals with the case of the Bell numbers. In contrast to Aitken's array A011971 and its variants A123346 and A011972, the Bell numbers do not appear as a column of the triangle but as row sums (times (-1)^n), i.e., as values of the associated polynomials at x = 1. Comparing this with a similar situation with the Bernoulli numbers/polynomials, our triangle could be viewed as a more organic generalization of the Bell numbers. Indeed, the names 'Bell triangle' and 'Bell polynomials' would be justified here; but these are already assigned to other concepts.

			The triangle T(n, k) starts:
  [0]     1;
  [1]    -2,      1;
  [2]     5,     -4,     1;
  [3]   -15,     15,    -6,      1;
  [4]    52,    -60,    30,     -8,    1;
  [5]  -203,    260,  -150,     50,  -10,    1;
  [6]   877,  -1218,   780,   -300,   75,  -12,   1;
  [7] -4140,   6139, -4263,   1820, -525,  105, -14,   1;
  [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;

Peter Luschny, Table of n, a(n) for row 0..100.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A293037 (row sums), A000110 (row sums, unsigned), A005493 (alternating row sums, signed).

Cf. A023531, A007318, A000079, A000012, A154921, A000670, A196838/A196839, A164555/A027642, A153641, A122045, A155585, A011971, A123346, A011972, A056857, A363524 (Chen sequence).

TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else
normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:
seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);

(* rows[0..n], n[0..oo] *)
(* row[n]= *)
n=9;r={};For[a=n+1,a>0,a--,AppendTo[r,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]];r
(* columns[1..n], n[0..oo] *)
(* column[n]= *)
n=0;c={};For[a=1,a<15,a++,AppendTo[c,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j-1))/(2*j),{j,1,n}]]];c
(* sequence *)
s={};For[n=0,n<15,n++,For[a=n+1,a>0,a--,AppendTo[s,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]]];s
(* Detlef Meya, Jun 22 2023 *)

def a(n): return (-1)^n / factorial(n)
@cached_function
def p(n, m):
    R = PolynomialRing(QQ, "x")
    if n == 0: return R(a(m))
    return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))
for n in range(10): print(p(n, 0).list())

A336589 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(1 - exp(x))).

Original entry on oeis.org

1, 0, -3, -19, -75, 574, 25795, 579963, 9342529, 21955076, -7954085799, -535479422655, -25206613635203, -871888114433454, -7465407495946777, 2538884115164554199, 344689220434285963905, 31689538033223254172648, 2273498459548301881979029

Offset: 0

Ilya Gutkovskiy, Jul 26 2020

sign

Cf. A001569, A009940, A293037, A336588.

nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[1 - Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
Table[n! Sum[(-1)^k StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]

a(n) = n! * sum(k=0, n, (-1)^k*stirling(n+1,k+1,2) / k!); \\ Michel Marcus, Jul 29 2020

a(n) = n! * Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) / k!.

A337062 E.g.f.: exp(1 + x^2/2 - exp(x)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 21, -51, 113, -498, 1088, -3335, 21407, -14653, 232389, -1275288, -3636526, -44468245, -7468609, 700603965, 12178055777, 67189448344, 175549544778, -2432123216941, -36279392911507, -287078642854853, -945866835928323

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

sign

Cf. A000587, A001147, A097514, A293037, A293038.

nmax = 26; CoefficientList[Series[Exp[1 + x^2/2 - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 26}]
Table[Sum[Binomial[n, 2 k] (2 k - 1)!! BellB[n - 2 k, -1], {k, 0, Floor[n/2]}], {n, 0, 26}]

a(0) = 1; a(n) = -a(n-1) - Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * (2*k-1)!! * A000587(n-2*k).

A337166 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 + x - BesselI(0,2*sqrt(x))).

Original entry on oeis.org

1, 0, -1, -1, 17, 99, -926, -20385, 25969, 7206059, 90298826, -3271747557, -149187119280, 236884125841, 233237751740057, 7110791842650002, -293292401726383791, -32980038867059802549, -498084376275585698222, 114298048468067933019627, 9072219653673352772098960

Offset: 0

Ilya Gutkovskiy, Jan 28 2021

sign

Cf. A061696, A293037, A336209.

A337166 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(binomial(n,k)^2*(n-k)*procname(k),k=0..n-2) ;
        -%/n ;
    end if;
    simplify(%) ;
end proc:
seq(A337166(n),n=0..40) ; # R. J. Mathar, Aug 19 2022

nmax = 20; CoefficientList[Series[Exp[1 + x - BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-2} binomial(n,k)^2 * (n-k) * a(k).

A332254 E.g.f.: 1 / (2 - exp(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 1, 1, 10, 31, 271, 1534, 14603, 120173, 1310224, 13947517, 175477699, 2265702388, 32673218085, 492565328493, 8053045395018, 138334722101571, 2535114408394699, 48790865853110950, 991843960201311455, 21121971129683138297, 471959969940724275432

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Cf. A000296, A028248, A083355, A293037.

nmax = 22; CoefficientList[Series[1/(2 - Exp[Exp[x] - 1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1/(2 - exp(exp(x + O(x*x^n)) - 1 - x))))} \\ Andrew Howroyd, Feb 08 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000296(k) * a(n-k).

a(n) ~ n! * exp(1 - exp(c-1)/2) / ((1 - 2*exp(1-c)) * (c - 1 - log(2))^(n+1)), where c = -LambertW(-1, -exp(-1)/2) = 2.678346990016660653412884512094523... - Vaclav Kotesovec, Feb 08 2020

A361531 Expansion of e.g.f. exp(1 - exp(x) + x^3/6).

Original entry on oeis.org

1, -1, 0, 2, -3, -2, 21, -44, -62, 631, -1367, -3170, 34849, -86855, -302964, 3058342, -8509971, -36488802, 430842051, -1111575888, -6244999438, 78663444549, -250850311489, -1724880111306, 18475299723737, -65061274823853, -444914618968648, 6831921081061986

Offset: 0

Seiichi Manyama, Mar 14 2023

Cf. A293037, A337062.

Cf. A124504, A361489.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-exp(x)+x^3/6)))

a(0) = 1, a(1) = -1, a(2) = 0; a(n) = binomial(n-1,2) * a(n-3) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A334192 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (k*j + 1)^n / ((-k)^j * j!).

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A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n*k + 1)^n/k!.

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A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

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A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

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A336589 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(1 - exp(x))).

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A337062 E.g.f.: exp(1 + x^2/2 - exp(x)).

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A337166 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 + x - BesselI(0,2*sqrt(x))).

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Maple

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A332254 E.g.f.: 1 / (2 - exp(exp(x) - 1 - x)).

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PARI

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A361531 Expansion of e.g.f. exp(1 - exp(x) + x^3/6).

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A334192 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (kj + 1)^n / ((-k)^j j!).

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k(nk + 1)^n/k!.