A225689 -id:A225689 - cp's OEIS frontend

A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = mT(n-1,k-1)+(k+1)T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 5, 8, 6, 6, 5, 16, 28, 40, 24, 24, 16, 61, 136, 180, 240, 120, 120, 61, 272, 662, 1232, 1320, 1680, 720, 720, 272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 1385, 7936, 24568, 56320, 83664, 129024, 100800, 120960, 40320, 40320, 7936, 50521, 176896, 408360, 814080, 1023120, 1491840, 1028160, 1209600, 362880, 362880

Offset: 0

N. J. A. Sloane, Mar 03 2022

Triangle connects Euler numbers on left and factorial numbers on right.

			Triangle begins:
    1,
    1,    1,
    1,    1,    1,
    1,    2,    2,    2,
    2,    5,    8,    6,     6,
    5,   16,   28,   40,    24,    24,
   16,   61,  136,  180,   240,   120,   120,
   61,  272,  662, 1232,  1320,  1680,   720,  720,
  272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040,
  ...
This may also be constructed as a square array, with entries T(n,k), n >= 1, 0 <= k, whose columns have e.g.f. equal to sec(x)+tan(x) (if k=0) and sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)) (if k>0):
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 5, 8, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, ...
5, 16, 28, 40, 24, 24, 0, 0, 0, 0, 0, 0, 0, ...
16, 61, 136, 180, 240, 120, 120, 0, 0, 0, 0, 0, 0, ...
61, 272, 662, 1232, 1320, 1680, 720, 720, 0, 0, 0, 0, 0, ...
272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 0, 0, 0, 0, ...
...

A. Boutin, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), 252-254.
E. Estanave, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), pp. 117-118.

Alois P. Heinz, Rows n = 0..140, flattened

The initial columns are A000111, A000111, A225689, A350971.
The diagonals, reading from the right, are (essentially) A000142, A000142, A002301, A006157, A002302, A350973, A002303, A350974, A350975.
Rows sums give A156142(n-1).
Cf. A007836.

for n from 0 to 12 do
T[n]:=Array(0..n,0);
T[0,0] := 1;
T[1,0] := 1; T[1,1] := 1;
if n>1 then
  T[n,0] := T[n-1,1];
for k from 1 to n-2 do
m:=k; if ((n+k) mod 2) = 0 then m:=k-1; fi;
T[n,k] := m*T[n-1,k-1] + (k+1)*T[n-1,k+1];
od:
T[n,n-1] := (n-1)*T[n-1,n-2];
T[n,n] := T[n,n-1];
fi;
lprint( [seq(T[n,k],k=0..n)] );
od:
# second Maple program:
b:= proc(n, i) option remember; `if`(i=0,
     `if`(n=0, 1, 0), b(n, i-1)+b(n-1, n-i))
    end:
T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k=0, b(n-1$2), `if`(n-k<=1, (n-1)!, (k+1)*
      T(n-1, k+1)+(k-irem(1+n+k, 2))*T(n-1, k-1))))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 04 2022
# To produce the square array, N. J. A. Sloane, Mar 05 2022:
read(transforms):
myegf := (f,M) -> SERIESTOLISTMULT(series(f,x,M));
T:=proc(n,k,M) local i;
if k=0 then myegf((sec(x)+tan(x)),M)[n];
else
myegf(sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)),M)[n];
fi;
end;
[seq(T(n,0,16),n=1..5)];
for n from 1 to 8 do
lprint([seq(T(n,k,16),k=0..12)]);
od:

b[n_, i_] := b[n, i] = If[i == 0,
     If[n == 0, 1, 0], b[n, i - 1] + b[n - 1, n - i]];
T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
     If[k == 0, b[n - 1, n - 1], If[n - k <= 1, (n - 1)!, (k + 1)*
     T[n - 1, k + 1] + (k - Mod[1 + n + k, 2])*T[n - 1, k - 1]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2022, Alois P. Heinz *)

If we ignore the n=0 row, then the e.g.f. for column 0 is sec(x)+tan(x), and for column k >= 1 it is sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)). See the initial rows of the square array in the EXAMPLES section. - N. J. A. Sloane, Mar 05 2022

abs(Sum_{k=0..n} (-1)^k * T(n,k)) = A007836(n) for n>=2. - Alois P. Heinz, Mar 04 2022

A225688 E.g.f.: sec(x)^3+(sec(x)^2*tan(x)).

Original entry on oeis.org

1, 1, 3, 8, 33, 136, 723, 3968, 25953, 176896, 1376643, 11184128, 101031873, 951878656, 9795436563, 104932671488, 1212135593793, 14544442556416, 186388033956483, 2475749026562048, 34859622790687713, 507711943253426176, 7791941518975112403, 123460740095103991808

Offset: 0

N. J. A. Sloane, May 26 2013

nonn

Number of up-down min-max permutations of n elements.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, 2013.
Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

Cf. A000111, A225689.

Table[n!*SeriesCoefficient[(1+Sin[x])/Cos[x]^3,{x,0,n}] ,{n,0,20}] (* Vaclav Kotesovec, May 26 2013 *)

The e.g.f. can also be written as (1+sin(x))/cos(x)^3.
a(n)+A225689(n) = A000111(n+2). - corrected by Vaclav Kotesovec, May 26 2013
a(n) ~ n! * n^2*(2/Pi)^(n+3). - Vaclav Kotesovec, May 26 2013

A164575 a(n) = n! * [x^n] 2(tan(x))^2(sec(x) + tan(x)).

Original entry on oeis.org

0, 0, 4, 12, 56, 240, 1324, 7392, 49136, 337920, 2652244, 21660672, 196658216, 1859020800, 19192151164, 206057828352, 2385488163296, 28669154426880, 367966308562084, 4893320282898432, 68978503204900376, 1005520890400604160, 15445185289163949004, 244890632417194278912

Offset: 0

Stefano Spezia, Aug 12 2019

Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

Cf. A000111, A000142, A000182, A000364, A009764, A013525, A024283, A181937, A225688, A225689.

gf := (2*sin(x)*tan(x))/(1 - sin(x)): ser := series(gf, x, 25):
seq(n!*coeff(ser, x, n), n=0..23); # Peter Luschny, Aug 19 2019

CoefficientList[Series[2Tan[x]^2(Sec[x]+Tan[x]),{x,0,23}],x]*Table[n!,{n,0,23}]

my(x='x+O('x^30)); concat([0,0], Vec(serlaplace(2*(tan(x))^2*(1/cos(x) + tan(x))))) \\ Michel Marcus, Aug 13 2019

a(n-2) = |{up-down 2nd-max-upper permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).
a(0) = 0 and a(n) = 2*A000142(n)*Sum_{i,j,k>=0, (2*i+1)+(2*j+1)+k=n} A000111(2*i+1)*A000111(2*j+1)*A000111(k)/(A000142(2*i+1)*A000142(2*j+1)*A000142(k)) for n > 0 (see Lemma 3.6 in Kobayashi).
a(2*n) = 2*A225689(2*n) (see Lemma 4.2 in Kobayashi).
a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - Vaclav Kotesovec, Aug 12 2019

A350971 Expansion of e.g.f.: (sec(x)tan(x))^2(1+sin(x)).

Original entry on oeis.org

0, 0, 2, 6, 40, 180, 1232, 7266, 56320, 408360, 3610112, 30974526, 309836800, 3065784540, 34342971392, 384653685786, 4778192404480, 59724464976720, 815553380483072, 11249503075325046, 167586815066767360, 2527964965265232900, 40815105402865713152, 668249973079223076306, 11626293107260590653440, 205304046476194041001080

Offset: 0

N. J. A. Sloane, Mar 03 2022

nonn

0, 0, followed by column 3 of A350970.

Cf. A000111, A225689, A350970.

E.g.f.: (sec(x)*tan(x))^2*(1+sin(x)).

A309845 Expansion of e.g.f.: sec(x) + 2*tan(x).

Original entry on oeis.org

1, 2, 1, 4, 5, 32, 61, 544, 1385, 15872, 50521, 707584, 2702765, 44736512, 199360981, 3807514624, 19391512145, 419730685952, 2404879675441, 58177770225664, 370371188237525, 9902996106248192, 69348874393137901, 2030847773013704704, 15514534163557086905, 493842960380415967232

Offset: 0

Stefano Spezia, Aug 20 2019

F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, 2013.
Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

Cf. A000111, A000182, A000364, A164575, A181937, A225688, A225689.

CoefficientList[Series[Sec[x]+2Tan[x],{x,0,25}],x]*Table[n!,{n,0,25}]

my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ Michel Marcus, Aug 20 2019

a(n-2) = |{up-down 2nd-max-lower permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).
a(n) = A000111(n+2) - A164575(n) (See Definition 3.4 in Kobayashi).
a(n) = A225688(n) + A225689(n) - A164575(n) (See Heneghan-Petersen and Kobayashi articles).
a(2*n) = A000111(2*n) (See Lemma 3.8 in Kobayashi).
a(2*n+1) = 2*A000111(2*n+1) (See Lemma 3.8 in Kobayashi).

cp's OEIS Frontend

A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = m*T(n-1,k-1)+(k+1)*T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

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A225688 E.g.f.: sec(x)^3+(sec(x)^2*tan(x)).

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Mathematica

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A164575 a(n) = n! * [x^n] 2*(tan(x))^2*(sec(x) + tan(x)).

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Programs

Maple

Mathematica

PARI

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A350971 Expansion of e.g.f.: (sec(x)*tan(x))^2*(1+sin(x)).

Views

Author

Keywords

Comments

Crossrefs

Formula

A309845 Expansion of e.g.f.: sec(x) + 2*tan(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = mT(n-1,k-1)+(k+1)T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

A164575 a(n) = n! * [x^n] 2(tan(x))^2(sec(x) + tan(x)).

A350971 Expansion of e.g.f.: (sec(x)tan(x))^2(1+sin(x)).