A000667 -id:A000667 - cp's OEIS frontend

A322268 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x)*(sec(x) + tan(x))^k.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 29, 24, 1, 1, 6, 25, 67, 105, 77, 1, 1, 7, 36, 129, 304, 433, 294, 1, 1, 8, 49, 221, 705, 1519, 2029, 1309, 1, 1, 9, 64, 349, 1416, 4145, 8386, 10709, 6664, 1, 1, 10, 81, 519, 2569, 9601, 26385, 51007, 63025, 38177, 1, 1, 11, 100, 737, 4320, 19777, 69406, 181969, 340024, 409713, 243034, 1

Offset: 0

Ilya Gutkovskiy, Dec 01 2018

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k + 1)^2*x^2/2! + (k^3 + 3*k^2 + 4*k + 1)*x^3/3! + (k^4 + 4*k^3 + 10*k^2 + 8*k + 1)*x^4/4! + ...
Square array begins:
  1,   1,    1,     1,     1,     1,  ...
  1,   2,    3,     4,     5,     6,  ...
  1,   4,    9,    16,    25,    36,  ...
  1,   9,   29,    67,   129,   221,  ...
  1,  24,  105,   304,   705,  1416,  ...
  1,  77,  433,  1519,  4145,  9601,  ...

Index entries for sequences related to boustrophedon transform

Columns k=0..3 give A000012, A000667, A292756, A292759.
Main diagonal gives A296793.
Cf. A322267.

Table[Function[k, n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(x)*(sec(x) + tan(x))^k.

A059234 The array in A059216 read by antidiagonals in the direction in which it was constructed.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 6, 10, 14, 1, 15, 26, 37, 45, 1, 46, 84, 121, 150, 169, 1, 170, 321, 471, 592, 686, 740, 1, 741, 1428, 2111, 2704, 3183, 3532, 3721, 1, 3722, 7255, 10777, 13953, 16685, 18826, 20347, 21142, 1, 21143, 41491, 61798, 80598, 97345, 111419

Offset: 1

Floor van Lamoen, Jan 18 2001

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

Cf. A000667, A059216, A059219, A059220, A059217.

More terms from Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A059237 Variation of Boustrophedon transform described in A059219 applied to sequence 0,1,0,0,0,....

Original entry on oeis.org

0, 1, 2, 5, 16, 59, 258, 1296, 7362, 46609, 325147, 2477212, 20460278, 182076531, 1736623109, 17672266151, 191111489038, 2188592796698, 26458831601847, 336735773968857, 4500142285227330, 63007188219787855, 922312862937555109

Offset: 0

N. J. A. Sloane, Jan 20 2001

nonn

Index entries for sequences related to boustrophedon transform

Cf. A000667, A059216, A059217, A059220, A059219.

aaa := proc(m,n) option remember; local i,j,r,s,t1; if m=0 and n=0 then RETURN(0); fi; if m=1 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1,n-1); for j from 0 to n-1 do t1 := t1+aaa(m,j); od: else t1 := aaa(m-1,n+1); for j from 0 to m-1 do t1 := t1+aaa(j,n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n,0), aaa(n-1,1), aaa(n-2,2), ..., aaa(0,n)

A059578 Variation of Boustrophedon transform applied to 1,1,1,1,... Fill an array by diagonals, all in the 'up' direction. The first column is 1,1,1,1,.... For the next element of a diagonal, add to the previous element the elements of the row and the column the new element is in. The first row gives a(n).

Original entry on oeis.org

1, 2, 7, 30, 147, 792, 4559, 27500, 171645, 1099388, 7185101, 47724494, 321225165, 2186177302, 15018795171, 104011496474, 725373340023, 5089785834004, 35907469451787, 254541483884544, 1812185157383017, 12951828431246472, 92893383046741073, 668383820775639066

Offset: 1

Floor van Lamoen, Jan 23 2001

			The array begins
1 2 7 30 ...
1 4 20 ...
1 8 ...
1 ...

Cf. A000667, A035002, A059216, A059219, A059502, A059513.

More terms from Sean A. Irvine, Sep 29 2022

A261880 Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.

Original entry on oeis.org

1, -1, -2, 1, 2, 4, -2, -3, -5, -9, 5, 7, 10, 15, 24, -16, -21, -28, -38, -53, -77, 61, 77, 98, 126, 164, 217, 294, -272, -333, -410, -508, -634, -798, -1015, -1309, 1385, 1657, 1990, 2400, 2908, 3542, 4340, 5355, 6664

Offset: 0

Paul Curtz, Jul 10 2016

Difference array of (-1)^n*A000111(n):
1, -1, 1, -2, 5, ...
-2, 2, -3, 7,...
4, -5, 10, ...
-9, 15, ...
24, ... .
First column:(-1)^n*A000667(n).
Antidiagonal sums: b(n) =  1, -3, 7, -19, 61, -233, 1037, -5279, 30241, ..., i.e., row sums of the triangle.
Any triangle with entries T(n, m) built from some sequence in column m=0, and the recurrence T(n, m) = T(n, m-1) - T(n-1, m-1) for m >= 1, has the property that the new triangle t(n, m) = T(n+1, m+1) - T(n+1, m), 0 <= m <= n, equals -T(n, m). See the question in the example. - Wolfdieter Lang, Aug 08 2016

			The triangle T(n, m) begins:
n\m  0   1   2   3   4   5 ...
0:   1
1:  -1  -2
2:   1   2   4
3:  -2  -3  -5  -9
4:   5   7  10  15  24,
5: -16 -21 -28 -38 -53 -77
...
Triangle of differences of the row entries of the preceding triangle starting with row n=1:
n\m  0   1    2   3   4 ...
0:  -1
1:   1   2
2:  -1  -2   -4
3:   2   3    5   9
4:  -5  -7  -10 -15 -24
... .
This is the negative of the first triangle. Are there other sequences with the same property?

Cf. A000111, A000667, A062162, A227862, A239005.

Recurrence: T(n, 0) = (-1)^n*A000111(n), n >= 0. T(n, m) = T(n, m-1) - T(n-1, m-1), m >= 1. (from the fact that the differences of the rows, starting with n = 1 produce the negative of the triangle. See the example and a comment). - Wolfdieter Lang, Aug 08 2016

Edited by Wolfdieter Lang, Aug 08 2016

A352906 Expansion of e.g.f. sinh(x) / (1 - sin(x)).

Original entry on oeis.org

0, 1, 2, 7, 24, 101, 472, 2507, 14784, 96361, 687392, 5332207, 44694144, 402663821, 3880880512, 39848805107, 434306095104, 5007757446481, 60907946680832, 779345606053207, 10465549612529664, 147168296199468341, 2162785172079204352, 33155700678534788507, 529311396083558989824

Offset: 0

Ilya Gutkovskiy, Apr 07 2022

nonn

Cf. A000111, A000667, A000795, A001250, A002084, A062161, A062272, A322623, A348580.

nmax = 24; CoefficientList[Series[Sinh[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * A000111(n-2*k).

a(n) ~ sinh(Pi/2) * 2^(n + 7/2) * n^(n + 3/2) / (exp(n) * Pi^(n + 3/2)). - Vaclav Kotesovec, Apr 07 2022

A363394 Triangle read by rows. T(n, k) = A081658(n, k) + A363393(n, k) for k > 0 and T(n, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, -1, 1, 3, -3, -2, 1, 4, -6, -8, 5, 1, 5, -10, -20, 25, 16, 1, 6, -15, -40, 75, 96, -61, 1, 7, -21, -70, 175, 336, -427, -272, 1, 8, -28, -112, 350, 896, -1708, -2176, 1385, 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936

Offset: 0

Peter Luschny, Jun 06 2023

			The triangle T(n, k) begins:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  -1;
  [3] 1, 3,  -3,   -2;
  [4] 1, 4,  -6,   -8,   5;
  [5] 1, 5, -10,  -20,  25,   16;
  [6] 1, 6, -15,  -40,  75,   96,   -61;
  [7] 1, 7, -21,  -70, 175,  336,  -427,  -272;
  [8] 1, 8, -28, -112, 350,  896, -1708, -2176,  1385;
  [9] 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936;

Richard P. Stanley, A survey of alternating permutations, arXiv:0912.4240 [math.CO], 2009.
Index entries for sequences related to Euler numbers.

Variants (row reversed): A109449, A247453.

Cf. A081658 (signed secant part), A363393 (signed tangent part), A000111 (main diagonal), A122045, A155585 (aerated main diagonal), A000667, A062162 (row sums of signless variant).

# Variant, computes abs(T(n, k)):
P := n -> n!*coeff(series((sec(y) + tan(y))/exp(x*y), y, 24), y, n):
seq(print(seq((-1)^(n - k)*coeff(P(n), x, n - k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k == n:
        p = k % 2
        return p - sum(T(n, j) for j in range(p, n - 1, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 1)])

|T(n, k)| = (-1)^(n - k) * n! * [x^(n - k)][y^n] (sec(y) + tan(y)) / exp(x*y).
T(n, k) = [x^(n - k)] -2^(k-(0^k))*(Euler(k, 0) + Euler(k, 1/2)) / (x-1)^(k + 1).
For a recursion see the Python program.
T(n, k) = [x^n] ((-1) + Sum_{j=0..n} binomial(n, j)*(Euler(j, 1) + Euler(j, 1/2))*(2*x)^j). - Peter Luschny, Nov 17 2024

cp's OEIS Frontend

A322268 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x)*(sec(x) + tan(x))^k.

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A059234 The array in A059216 read by antidiagonals in the direction in which it was constructed.

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A059237 Variation of Boustrophedon transform described in A059219 applied to sequence 0,1,0,0,0,....

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Maple

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A261880 Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.

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Comments

Examples

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Formula

Extensions

A352906 Expansion of e.g.f. sinh(x) / (1 - sin(x)).

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Mathematica

Formula

A363394 Triangle read by rows. T(n, k) = A081658(n, k) + A363393(n, k) for k > 0 and T(n, 0) = 1.

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