A373428 -id:A373428 - 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

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.

A373427 Triangle read by rows: Coefficients of the polynomials SC(n, x) * EZ(n, x), where SC denote the Stirling cycle polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 6, 29, 45, 30, 9, 1, 0, 24, 218, 553, 629, 366, 112, 17, 1, 0, 120, 1954, 7781, 13409, 12136, 6270, 1894, 326, 29, 1, 0, 720, 20484, 125968, 313715, 407297, 308286, 143725, 42124, 7683, 830, 47, 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,   5,   4,   1]
4: [1, 3, 1] *    [0,  6, 11, 6, 1] =      [0,  6,  29,  45,  30,   9,   1]
5: [1, 7, 7, 1] * [0, 24, 50, 35, 10, 1] = [0, 24, 218, 553, 629, 366, 112,17,1]

Peter Luschny, Illustrating the polynomials.

Cf. A132393 (Stirling cycle), A205497 (zig-zag Eulerian), A373433 (row sums).

Cf. A373429, A373428, A373432.

EZP((n, k) -> abs(Stirling1(n, k)), 7);  # Using function EZP from A373432.

A373571 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EP(n, x), where S2 denote the Stirling set polynomials and EP the Eulerian polynomials A173018.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 1, 7, 14, 7, 1, 0, 1, 18, 94, 145, 84, 17, 1, 0, 1, 41, 481, 1676, 2302, 1351, 351, 36, 1, 0, 1, 88, 2159, 14859, 40319, 49434, 29378, 8627, 1222, 72, 1, 0, 1, 183, 9052, 113919, 554030, 1236040, 1380913, 816404, 260968, 44577, 3851, 141, 1

Offset: 0

Peter Luschny, Jun 15 2024

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 1,  2,    1]
[3] [0, 1,  7,   14,     7,     1]
[4] [0, 1, 18,   94,   145,    84,    17,     1]
[5] [0, 1, 41,  481,  1676,  2302,  1351,   351,   36,    1]
[6] [0, 1, 88, 2159, 14859, 40319, 49434, 29378, 8627, 1222, 72, 1]

Cf. A048993, A173018, A373657, A373428, A320956, A137341 (row sums).

PolyProd(Stirling2, combinat:-eulerian1, 7);  # Using PolyProd from A373657.

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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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.

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A373427 Triangle read by rows: Coefficients of the polynomials SC(n, x) * EZ(n, x), where SC denote the Stirling cycle polynomials and EZ the Eulerian zig-zag polynomials A205497.

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A373571 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EP(n, x), where S2 denote the Stirling set polynomials and EP the Eulerian polynomials A173018.

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