A094416 -id:A094416 - cp's OEIS frontend

A320080 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - k*log(1 + x)).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 2, 0, 1, 4, 15, 28, 4, 0, 1, 5, 28, 114, 172, 14, 0, 1, 6, 45, 296, 1152, 1328, 38, 0, 1, 7, 66, 610, 4168, 14562, 12272, 216, 0, 1, 8, 91, 1092, 11020, 73376, 220842, 132480, 600, 0, 1, 9, 120, 1778, 24084, 248870, 1550048, 3907656, 1633344, 6240, 0

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k - 1)*x^2/2! + 2*k*(3*k^2 - 3*k + 1)*x^3/3! + 2*k*(12*k^3 - 18*k^2 + 11*k - 3)*x^4/4! + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   1,     6,     15,     28,      45,  ...
  0,   2,    28,    114,    296,     610,  ...
  0,   4,   172,   1152,   4168,   11020,  ...
  0,  14,  1328,  14562,  73376,  248870,  ...

Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.
Main diagonal gives A317172.
Cf. A048594, A048994, A094416, A320079, A334369.

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

E.g.f. of column k: 1/(1 - k*log(1 + x)).
A(n,k) = Sum_{j=0..n} Stirling1(n,j)*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A238464 Generalized ordered Bell numbers Bo(7,n).

Original entry on oeis.org

1, 7, 105, 2359, 70665, 2646007, 118893705, 6232661239, 373405001865, 25167452766967, 1884759251911305, 155262005162499319, 13952854271421949065, 1358385484966283220727, 142418920493123648992905, 15998363870912950298468599

Offset: 0

Vincenzo Librandi, Mar 17 2014

nonn

Row 7 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(8 - 7*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

t=30; Range[0, t]! CoefficientList[Series[1/(8 - 7 Exp[x]), {x, 0, t}], x]

x='x+O('x^66); Vec(serlaplace(1/(8 - 7*exp(x)))) \\ Joerg Arndt, Mar 17 2014

E.g.f.: 1/(8 - 7*exp(x)).
a(n) ~ n! / (8*(log(8/7))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 7*a(n-1) - 8*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  0,    1,     2,      3,       4,       5,  ...
  0,    3,    10,     21,      36,      55,  ...
  0,   14,    76,    222,     488,     910,  ...
  0,   88,   772,   3132,    8824,   20080,  ...
  0,  694,  9808,  55242,  199456,  553870,  ...

Columns k=0..5 give A000007, A007840, A088500, A354263, A354264, A365588.
Main diagonal gives A317171.
Cf. A048594, A048994, A094416, A320080.

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

E.g.f. of column k: 1/(1 + k*log(1 - x)).
A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset: 0

Peter Luschny, May 21 2021

The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - Peter Luschny, Apr 27 2024

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
.
Seen as an array A(n, k) = T(n + k, n):
[0] [1, 0,   0,    0,     0,       0,         0, ...  A000007
[1] [1, 1,   3,   13,    75,     541,      4683, ...  A000670
[2] [1, 2,  10,   74,   730,    9002,    133210, ...  A004123
[3] [1, 3,  21,  219,  3045,   52923,   1103781, ...  A032033
[4] [1, 4,  36,  484,  8676,  194404,   5227236, ...  A094417
[5] [1, 5,  55,  905, 19855,  544505,  17919055, ...  A094418
[6] [1, 6,  78, 1518, 39390, 1277646,  49729758, ...  A094419
[7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ...  A238464

Variant of the array is A094416 (which has column 0 and row 0 missing).
The coefficients of the Fubini polynomials are A131689.
Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

F[n_] := F[n] = If[n == 0, 1,
   Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 06 2024, after Peter Luschny *)

# Computes the triangle.
@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

# Computes the square array using the Akiyama-Tanigawa algorithm.
def ATFubini(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = (n + 1)**k  # Chancing this to R[k] = k**n generates A371761.
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(8): print([n], ATFubini(n, 7))  # Peter Luschny, Apr 27 2024

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

A238465 Generalized ordered Bell numbers Bo(8,n).

Original entry on oeis.org

1, 8, 136, 3464, 117640, 4993928, 254396296, 15119104904, 1026912225160, 78468091562888, 6662087721342856, 622186077361470344, 63389713864392140680, 6996476832548305415048, 831619554631233264449416, 105909083171031626820475784

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 8 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(9 - 8*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t = 30; Range[0, t]! CoefficientList[Series[1/(9 - 8 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(9 - 8*exp(x)).
a(n) ~ n! / (9*(log(9/8))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 8*a(n-1) - 9*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A094421 a(n) = n * (6n^2 + 6n + 1).

Original entry on oeis.org

13, 74, 219, 484, 905, 1518, 2359, 3464, 4869, 6610, 8723, 11244, 14209, 17654, 21615, 26128, 31229, 36954, 43339, 50420, 58233, 66814, 76199, 86424, 97525, 109538, 122499, 136444, 151409, 167430, 184543, 202784, 222189, 242794

Offset: 1

Ralf Stephan, May 02 2004

Third column of array A094416.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

List([1..40], n-> n*(6*n^2+6*n+1)); # G. C. Greubel, Oct 30 2019

[n*(6*n^2+6*n+1): n in [1.40]]; // G. C. Greubel, Oct 30 2019

A094421:=n->n * (6*n^2 + 6*n + 1); seq(A094421(n), n=1..40); # Wesley Ivan Hurt, Feb 12 2014

Table[n(6n^2+6n+1), {n, 40}] (* Wesley Ivan Hurt, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{13,74,219,484},40] (* Harvey P. Dale, Jan 04 2016 *)

vector(40, n, n*(6*n^2+6*n+1)) \\ G. C. Greubel, Oct 30 2019

[n*(6*n^2+6*n+1) for n in (1..40)] # G. C. Greubel, Oct 30 2019

Equals n * A003154(n) (star numbers).
G.f.: x*(13 + 22*x + x^2)/(1-x)^4. - Colin Barker, Aug 02 2012
E.g.f.: x*(13 + 24*x + 6*x^2)*exp(x). - G. C. Greubel, Oct 30 2019

A238466 Generalized ordered Bell numbers Bo(9,n).

Original entry on oeis.org

1, 9, 171, 4869, 184851, 8772309, 499559571, 33190014069, 2520110222451, 215270320769109, 20431783142389971, 2133148392099721269, 242954208655633344051, 29977118969127060357909, 3983272698956314883956371, 567091857051921058649396469

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 9 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(10 - 9*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(10 - 9 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(10 - 9*exp(x)).
a(n) ~ n! / (10*(log(10/9))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 9*a(n-1) - 10*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A238467 Generalized ordered Bell numbers Bo(10,n).

Original entry on oeis.org

1, 10, 210, 6610, 277410, 14553010, 916146210, 67285818610, 5647734061410, 533307215001010, 55954905981282210, 6457903731351210610, 813080459351919805410, 110901542660769629769010, 16290196917457939734258210

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 10 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(11 - 10*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(11 - 10 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(11 - 10*exp(x)).
a(n) ~ n! / (11*(log(11/10))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 10*a(n-1) - 11*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

cp's OEIS Frontend

A320080 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - k*log(1 + x)).

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A238464 Generalized ordered Bell numbers Bo(7,n).

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A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

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A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A238465 Generalized ordered Bell numbers Bo(8,n).

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A094421 a(n) = n * (6*n^2 + 6*n + 1).

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A238466 Generalized ordered Bell numbers Bo(9,n).

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Magma

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A094421 a(n) = n * (6n^2 + 6n + 1).