A320956 -id:A320956 - cp's OEIS frontend

A321400 A family of sequences converging to the exponential limit of sec + tan (A320956). Square array A(n, k) for n >= 0 and k >= 0, read by descending antidiagonals.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 5, 8, 2, 1, 1, 0, 16, 40, 10, 2, 1, 1, 0, 61, 256, 70, 10, 2, 1, 1, 0, 272, 1952, 656, 75, 10, 2, 1, 1, 0, 1385, 17408, 7442, 816, 75, 10, 2, 1, 1, 0, 7936, 177280, 99280, 11407, 832, 75, 10, 2, 1, 1

Offset: 0

Peter Luschny, Nov 08 2018

See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.

			Array starts:
n\k   0  1  2   3   4    5      6       7        8  ...
-------------------------------------------------------
  [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
  [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
  [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
  [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
  [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
  [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
  [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
  [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
  [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
-------------------------------------------------------
Seen as a triangle given by descending antidiagonals:
  [0] 1
  [1] 0,  1
  [2] 0,  1,   1
  [3] 0,  1,   1,  1
  [4] 0,  2,   2,  1,  1
  [5] 0,  5,   8,  2,  1, 1
  [6] 0, 16,  40, 10,  2, 1, 1
  [7] 0, 61, 256, 70, 10, 2, 1, 1

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4), A320956 (limit).

Antidiagonal sums (and row sums of the triangle): A321399.

sf := proc(n) option remember; `if`(n <= 1, 1-n, (n-1)*(sf(n-1) + sf(n-2))) end:
kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
egf := n -> add(kernel(n, k)*((tan + sec)(x*(n - k))), k=0..n):
A321400Row := proc(n, len) series(egf(n), x, len + 2):
seq(coeff(%, x, k)*k!/n!, k=0..len) end:
seq(lprint(A321400Row(n, 9)), n=0..9);

A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 8, 1, 0, 1, 1, 2, 5, 14, 16, 1, 0, 1, 1, 2, 5, 15, 41, 32, 1, 0, 1, 1, 2, 5, 15, 51, 122, 64, 1, 0, 1, 1, 2, 5, 15, 52, 187, 365, 128, 1, 0, 1, 1, 2, 5, 15, 52, 202, 715, 1094, 256, 1, 0

Offset: 0

Peter Luschny, Nov 05 2018

Arndt and Sloane (see the link and A278984) identify the sequence to give "the number of words of length n over an alphabet of size b that are in standard order" and provide the formula Sum_{j = 1..b} Stirling_2(n, j) assuming b >= 1 and j >= 1. Compared to the array as defined here this misses the first row and the first column of our array.
The method used here is the special case of a general method described in A320956 applied to the function exp. For applications to other functions see the cross references.
A(k,n) is the number of color patterns (set partitions) for an oriented row of length n using up to k colors (subsets). Two color patterns are equivalent if the colors are permuted. For A(3,4) = 14, the six achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA; the eight chiral patterns are the four chiral pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Nov 10 2018

			Array starts:
n\k   0  1  2  3   4   5    6    7     8      9  ...
----------------------------------------------------
[0]   1, 0, 0, 0,  0,  0,   0,   0,    0,     0, ...  A000007
[1]   1, 1, 1, 1,  1,  1,   1,   1,    1,     1, ...  A000012
[2]   1, 1, 2, 4,  8, 16,  32,  64,  128,   256, ...  A011782
[3]   1, 1, 2, 5, 14, 41, 122, 365, 1094,  3281, ...  A124302
[4]   1, 1, 2, 5, 15, 51, 187, 715, 2795, 11051, ...  A124303
[5]   1, 1, 2, 5, 15, 52, 202, 855, 3845, 18002, ...  A056272
[6]   1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, ...  A056273, ?A284727
[7]   1, 1, 2, 5, 15, 52, 203, 877, 4139, 21110, ...
[8]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, ...
[9]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
----------------------------------------------------
Seen as a triangle given by the descending antidiagonals:
[0]             1
[1]            0, 1
[2]          0, 1, 1
[3]        0, 1, 1, 1
[4]       0, 1, 2, 1, 1
[5]     0, 1, 4, 2, 1, 1
[6]    0, 1, 8, 5, 2, 1, 1
[7]  0, 1, 16, 14, 5, 2, 1, 1

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Cf. A008290, A000110, A137650, A211561, A000240, A000387, A000449, A103816/A053556.
Cf. A124302, A124303, A056272, A056273, A284727, A278984.
Antidiagonal sums (and row sums of the triangle): A320964.
Cf. this sequence (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), A320959 (arctanh).
Cf. A320750 (unoriented), A320751 (chiral), A305749 (achiral).

A := (n, k) -> if k = 0 then 1 else add(A008290(n, n-j)*(n-j)^k, j=0..n-1)/n! fi:
seq(lprint(seq(A(n, k), k=0..9)), n=0..9); # Prints the array row-wise.
seq(seq(A(n-k, k), k=0..n), n=0..11); # Gives the array as listed.

T[n_, 0] := 1; T[n_, k_] := Sum[(Subfactorial[j]/Factorial[j])((n - j)^k/(n - j)!), {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten
Table[Sum[StirlingS2[k, j], {j, 0, n-k}], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert A. Russell, Nov 10 2018 *)

A(n, k) = (1/n!)*Sum_{j=0..n-1} A008290(n, n-j)*(n-j)^k if k > 0.
If one drops the special case A(n, 0) = 1 from the definition then column 0 becomes Sum_{k=0..n} (-1)^k/k! = A103816(n)/A053556(n).
Row n is given for k >= 1 by a_n(k), where
a_0(k) = 0^k/0!.
a_1(k) = 1^k/1!.
a_2(k) = (2^k)/2!.
a_3(k) = (3^k + 3)/3!.
a_4(k) = (6*2^k + 4^k + 8)/4!.
a_5(k) = (20*2^k + 10*3^k + 5^k + 45)/5!.
a_6(k) = (135*2^k + 40*3^k + 15*4^k + 6^k + 264)/6!.
a_7(k) = (924*2^k + 315*3^k + 70*4^k + 21*5^k + 7^k + 1855)/7!.
a_8(k) = (7420*2^k + 2464*3^k + 630*4^k + 112*5^k + 28*6^k + 8^k + 14832)/8!.
Note that the coefficients of the generating functions a_n are the recontres numbers A000240, A000387, A000449, ...
Rewriting the formulas with exponential generating functions for the rows we have egf(n) = Sum_{k=0..n} !k*binomial(n,k)*exp(x*(n-k)) and A(n, k) = (k!/n!)*[x^k] egf(n). In this formulation no special rule for the case k = 0 is needed.
The rows converge to the Bell numbers. Convergence here means that for every fixed k the terms in column k differ from A000110(k) only for finitely many indices.
A(n, n) are the Bell numbers A000110(n) for n >= 0.
Let S(n, k) = Bell(n+k+1) - A(n, k+n+1) for n >= 0 and k >= 0, then the square array S(n, k) read by descending antidiagonals equals provable the triangle A137650 and equals empirical the transpose of the array A211561.

A320962 a(n) = (-1)^(n-1)(n-1)!Sum_{i=0..n} Stirling2(n, i) if n > 0 and 0 otherwise.

Original entry on oeis.org

0, 1, -2, 10, -90, 1248, -24360, 631440, -20865600, 852647040, -42085008000, 2462394816000, -168193308729600, 13241729554099200, -1188734048799897600, 120563962753538304000, -13704613258628388096000, 1733764260005567741952000, -242606144946628642443264000

Offset: 0

Peter Luschny, Nov 07 2018

sign

Also the exponential limit as defined in A320956 of log(x + 1).

			Illustration of the convergence in the sense of A320956:
  [0] 0, 0,  0,  0,   0,    0,      0,      0,         0, ...
  [1] 0, 1, -1,  2,  -6,   24,   -120,    720,     -5040, ... A133942
  [2] 0, 1, -2,  8, -48,  384,  -3840,  46080,   -645120, ... A000165
  [3] 0, 1, -2, 10, -84,  984, -14640, 262800,  -5513760, ... A321398
  [4] 0, 1, -2, 10, -90, 1224, -22440, 514800, -14086800, ...
  [5] 0, 1, -2, 10, -90, 1248, -24240, 615600, -19378800, ...
  [6] 0, 1, -2, 10, -90, 1248, -24360, 630720, -20719440, ...
  [7] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20860560, ...
  [8] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20865600, ...

Cf. A320956, A133942 (n=1), A000165 (n=2), A321398 (n=3).

a := n -> `if`(n=0, 0, (-1)^(n-1)*(n-1)!*add(Stirling2(n, i), i=0..n)):
seq(a(n), n=0..19);
# Alternatively use the function ExpLim defined in A320956.
ExpLim(19, x -> ln(x+1));

a[n_] := If[n == 0, 0, (-1)^(n - 1)*(n - 1)!*Sum[StirlingS2[n, i], {i, 0, n}]]; Array[a, 19, 0] (* Amiram Eldar, Nov 07 2018 *)

a(n) = if (n>0, (-1)^(n-1)*(n-1)!*sum(i=0, n, stirling(n, i, 2)), 0); \\ Michel Marcus, Nov 07 2018

A373428 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EZ(n, x), where S2 denote the Stirling set polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 4, 1, 0, 1, 10, 28, 26, 9, 1, 0, 1, 22, 137, 291, 261, 102, 17, 1, 0, 1, 45, 555, 2300, 4150, 3517, 1479, 306, 29, 1, 0, 1, 89, 2048, 15152, 48942, 76259, 61846, 26976, 6388, 795, 47, 1

Offset: 0

Peter Luschny, Jun 06 2024

			Tracing the computation:
0: [1] *          [1] =                   [1]
1: [1] *          [0, 1] =                [0, 1]
2: [1] *          [0, 1, 1] =             [0, 1, 1]
3: [1, 1] *       [0, 1, 3, 1] =          [0, 1, 4, 4, 1]
4: [1, 3, 1] *    [0, 1, 7, 6, 1] =       [0, 1, 10, 28, 26, 9, 1]
5: [1, 7, 7, 1] * [0, 1, 15, 25, 10, 1] = [0, 1, 22, 137, 291, 261, 102, 17, 1]

Peter Luschny, Illustrating the polynomials.

Cf. A048993 (Stirling2), A205497 (zig-zag Eulerian), A320956 (row sums).

EZP(Stirling2, 7);  # Using function EZP from A373432.

A320958 The exponential limit of arcsin (odd indices only).

Original entry on oeis.org

1, 5, 468, 197325, 233145675, 605979974250, 2987147975582925, 25254853526009732625, 340477692051264295027500, 6926101229658271208893970625, 203562520854789108487169894574375, 8346651541805126492397454664310896250, 463877742240727904202821053051014479795625

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for definitions and comments.

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0, 0, 0,   0, 0,      0, 0,         0, ...
   [1] 0, 1, 0, 1, 0,   9, 0,    225, 0,     11025, ... A177145, A001818
   [2] 0, 1, 0, 4, 0, 144, 0,  14400, 0,   2822400, ... A122747
   [3] 0, 1, 0, 5, 0, 369, 0,  82125, 0,  36173025, ...
   [4] 0, 1, 0, 5, 0, 459, 0, 160875, 0, 121837275, ...
   [5] 0, 1, 0, 5, 0, 468, 0, 192375, 0, 198472050, ...
   [6] 0, 1, 0, 5, 0, 468, 0, 197100, 0, 227644200, ...
   [7] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 232737750, ...
   [8] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233134650, ...
   [9] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233145675, ...

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), this sequence (arcsin),
A320959 (arctanh).
Cf. A177145, A001818, A122747.

# Function ExpLim defined in A320956.
L := [ExpLim(28, arcsin)]: seq(L[2*n], n=1..13);

m = 13; CoefficientList[ArcSin[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

A320959 The exponential limit of arctanh (odd indices only).

Original entry on oeis.org

1, 10, 1248, 631440, 852647040, 2462394816000, 13241729554099200, 120563962753538304000, 1733764260005567741952000, 37343395325946891151466496000, 1155311729350231354981936496640000, 49626886713956000390638096497377280000, 2878005957927359237424925417166882734080000

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for definitions and comments.

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0,  0, 0,    0, 0,      0, 0,         0, ...
   [1] 0, 1, 0,  2, 0,   24, 0,    720, 0,     40320, ... A005359
   [2] 0, 1, 0,  8, 0,  384, 0,  46080, 0,  10321920, ... A067624
   [3] 0, 1, 0, 10, 0,  984, 0, 262800, 0, 132289920, ...
   [4] 0, 1, 0, 10, 0, 1224, 0, 514800, 0, 445576320, ...
   [5] 0, 1, 0, 10, 0, 1248, 0, 615600, 0, 725840640, ...
   [6] 0, 1, 0, 10, 0, 1248, 0, 630720, 0, 832527360, ...
   [7] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 851155200, ...
   [8] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852606720, ...
   [9] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852647040, ...

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), this sequence (arctanh).

Cf. A005359, A067624.

# The function ExpLim is defined in A320956.
L := [ExpLim(28, arctanh)]: seq(L[2*n], n=1..13);

m = 13; CoefficientList[ArcTanh[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

For n >= 3 and odd, -a(m)*Zeta(m) = g(n), where g denotes the exponential limit of log(Gamma(x + 1)) and m = (n-1)/2.

A320957 a(n) = (1/6)n![x^n] (2 + sec(3x) + tan(3x) + 3sec(x) + 3tan(x)).

Original entry on oeis.org

1, 1, 2, 10, 70, 656, 7442, 99280, 1515190, 26038016, 497227682, 10445708800, 239394707110, 5943715352576, 158922998335922, 4552807055288320, 139123511874743830, 4517007538261262336, 155283277843358756162, 5634815061983539363840, 215234080472925069593350

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for motivation and definitions.

Cf. A000111 (n=1), A000828 (n=2), this sequence (n=3), A321394 (n=4), A320956.

egf := 2 + sec(3*x) + tan(3*x) + 3*sec(x) + 3*tan(x):
ser := series(egf, x, 22): seq((1/6)*n!*coeff(ser, x, n), n=0..20);

m = 20;
egf = 2 + Sec[3x] + Tan[3x] + 3 Sec[x] + 3 Tan[x];
(1/6) CoefficientList[egf + O[x]^(m+1), x] Range[0, m]! (* Jean-François Alcover, Aug 19 2021 *)

sectan(x) = 1/cos(x) + tan(x);
my(x='x+O('x^25)); Vec(serlaplace(2 + sectan(3*x) + 3*sectan(x)))/6 \\ Michel Marcus, Aug 19 2021

A321394 a(n) = (1/24)n![x^n] (9 + sectan(4x) + 6sectan(2x) + 8sectan(x)) where sectan(x) = sec(x) + tan(x).

Original entry on oeis.org

1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, 87700736, 2220246387, 62010892800, 1892138207375, 62591994720256, 2230631475837767, 85188256574494720, 3470563987113896475, 150234341045137637376, 6886077311552162511547, 333165973379285030666240, 16967906593223743786978375

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for motivation and definitions.

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), this sequence (n=4), A320956.

sectan := x -> sec(x) + tan(x): # sin(Pi/4 + x/2)*csc(Pi/4 - x/2)
egf := 9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x):
ser := series(egf, x, 22): seq((1/24)*n!*coeff(ser, x, n), n=0..20);

m = 20;
sectan[x_] := Sec[x] + Tan[x];
egf = 9 + sectan[4x] + 6 sectan[2x] + 8 sectan[x];
(1/24) CoefficientList[egf + O[x]^(m+1), x] Range[0, m]! (* Jean-François Alcover, Aug 19 2021 *)

sectan(x) = 1/cos(x) + tan(x);
my(x='x+O('x^25)); Vec(serlaplace(9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)))/24 \\ Michel Marcus, Aug 19 2021

A373429 Triangle read by rows: Coefficients of the polynomials S1(n, x) * EZ(n, x), where S1 denote the Stirling1 polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -1, -2, 1, 0, -6, -7, 21, -6, -3, 1, 0, 24, 118, -147, -91, 126, -28, -3, 1, 0, -120, -1406, -109, 3749, -2084, -450, 514, -94, -1, 1, 0, 720, 16956, 34240, -72307, -15475, 56286, -21125, -674, 1635, -262, 5, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
0: [1] *          [1] =                      [1]
1: [1] *          [0,  1] =                  [0,  1]
2: [1] *          [0, -1,  1] =              [0, -1,   1]
3: [1, 1] *       [0,  2,  -3,  1] =         [0,  2,  -1,   -2,   1]
4: [1, 3, 1] *    [0, -6,  11, -6,   1] =    [0, -6,  -7,   21,  -6,  -3,  1]
5: [1, 7, 7, 1] * [0, 24, -50, 35, -10, 1] = [0, 24, 118, -147, -91, 126,-28,-3,1]

Peter Luschny, Illustrating the polynomials.

Cf. A048994 (Stirling1), A205497 (zig-zag Eulerian), A320956 (row sums).

Cf. A373428, A373432.

EZP(Stirling1, 7);  # Using function EZP from A373432.

A320961 The exponential limit of (-x)!, rounded to the nearest integer.

Original entry on oeis.org

1, 1, 4, 27, 353, 6128, 145159, 4402407, 166608593, 7666343436, 420646243820, 27079750092637, 2018074017351900, 172131994564410026, 16641769389384512884, 1808431867178308597550, 219272140061011055068448, 29473880023661693302772550, 4366902281695075479226089449

Offset: 0

Peter Luschny, Nov 07 2018

nonn

The exponential limit of a function is defined in A320956. Applied to (-x)! Maple returns a sequence of sums of Zeta values, powers of Pi, powers of Euler's gamma, etc.. The sequence starts: 1, gamma, (1/3)*Pi^2 + 2* gamma^2, 10*Zeta(3) + (5/2)*Pi^2*gamma + 5*gamma^3, ... These sums, rounded to the nearest integer, give the sequence.

Cf. A320956, A000142.

The exponential limit of other functions: A320955 (exp), A320962 (log(x+1)), A320958 (arcsin), A320959 (arctanh).

explim := (len, f) -> seq(combinat:-bell(n)*((D@@n)(f))(0), n=0..len):
explim(18, x -> (-x)!): map(round, [evalf(%, 46)]);

m = 19; CoefficientList[(-x)!+O[x]^m, x]*Range[0, m-1]!*BellB[Range[0, m-1]] // Round (* Jean-François Alcover, Jul 21 2019 *)

cp's OEIS Frontend

A321400 A family of sequences converging to the exponential limit of sec + tan (A320956). Square array A(n, k) for n >= 0 and k >= 0, read by descending antidiagonals.

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A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

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Formula

A320962 a(n) = (-1)^(n-1)*(n-1)!*Sum_{i=0..n} Stirling2(n, i) if n > 0 and 0 otherwise.

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PARI

A373428 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EZ(n, x), where S2 denote the Stirling set polynomials and EZ the Eulerian zig-zag polynomials A205497.

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A320958 The exponential limit of arcsin (odd indices only).

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A320959 The exponential limit of arctanh (odd indices only).

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A320957 a(n) = (1/6)*n!*[x^n] (2 + sec(3*x) + tan(3*x) + 3*sec(x) + 3*tan(x)).

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A321394 a(n) = (1/24)*n!*[x^n] (9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)) where sectan(x) = sec(x) + tan(x).

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A320962 a(n) = (-1)^(n-1)(n-1)!Sum_{i=0..n} Stirling2(n, i) if n > 0 and 0 otherwise.

A320957 a(n) = (1/6)n![x^n] (2 + sec(3x) + tan(3x) + 3sec(x) + 3tan(x)).

A321394 a(n) = (1/24)n![x^n] (9 + sectan(4x) + 6sectan(2x) + 8sectan(x)) where sectan(x) = sec(x) + tan(x).