A320957 -id:A320957 - cp's OEIS frontend

A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, 167822592, 5859172975, 240072637440, 11388362495705, 618357843791872, 38057876106154882, 2632817442236631040, 203225803724876875315, 17390464322078045896704, 1640312648221489789841119, 169667967895669459925991424

Offset: 0

Peter Luschny, Nov 07 2018

nonn

We say that the sequence S is the exponential limit of the function f relative to the kernel K if and only if the exponential generating functions
  egf(n) = Sum_{k=0..n} K(n, k)*f(x*(n-k)) generate a family of sequences
T(n) = k -> (k!/n!)*[x^k] egf(n) which converge to S. Convergence here means that for every fixed k the terms T(n)(k) differ from S(k) only for finitely many indices.
The paradigmatic example is to set f(x) = exp(x), K(n, k) = !k*binomial(n, k) (!n is the subfactorial of n) and obtain for S the Bell numbers. This example is set forth in A320955.
Let D(f)(x) represent the derivative of f(x) with respect to x and (D^(n))(f) the n-th derivative of f. Then the exponential limit of f is B(n)*((D^(n))(f))(0) where B(n) is the n-th Bell number: ExpLim(f) = f(0), (D(f))(0), 2*((D^(2))(f))(0), 5*((D^(3))(f))(0), 15*((D^(4))(f))(0), 52*((D^(5))(f))(0), ... Since exp is a fixed point of D and exp(0) = 1 we have the identity ExpLim(exp)[n] = B(n). Similarly ExpLim(sin)[n] = B(n)*mod(n,2)*(-1)^binomial(n,2).
If we set f = sec + tan and K(n, k) = !k*binomial(n, k) the exponential limit is this sequence, a(n).

			Illustration of the convergence:
  [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, ...

Peter Luschny, Table of n, a(n) for n = 0..296

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4).
Cf. A320955 (exp), A320962 (log(x+1)), this sequence (sec+tan), A320958 (arcsin), A320959 (arctanh).
Cf. A373428.

ExpLim := proc(len, f) local kernel, sf, egf:
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)*f(x*(n-k)), k=0..n):
series(egf(len), x, len+2): seq(coeff(%, x, k)*k!/len!, k=0..len) end:
ExpLim(19, sec + tan);
# Alternative:
explim := (len, f) -> seq(combinat:-bell(n)*((D@@n)(f))(0), n=0..len):
explim(19, sec + tan);
# Or:
a := n -> A000110(n)*A000111(n): seq(a(n), n = 0..19);  # Peter Luschny, Jun 07 2024

m = 20; CoefficientList[Sec[x] + Tan[x] + O[x]^m, x] * Range[0, m-1]! *
BellB[Range[0, m-1]] (* Jean-François Alcover, Jun 19 2019 *)

Name extended by Peter Luschny, Jun 07 2024

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

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.

Original entry on oeis.org

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);

cp's OEIS Frontend

A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

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

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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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Maple

cp's OEIS Frontend

A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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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Author

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Comments

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Maple

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

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