A075497 -id:A075497 - cp's OEIS frontend

A016283 a(n) = 6^n/8 - 4^(n-1) + 2^(n-3).

0, 0, 1, 12, 100, 720, 4816, 30912, 193600, 1194240, 7296256, 44301312, 267904000, 1615810560, 9728413696, 58504691712, 351565004800, 2111537479680, 12677814747136, 76101248090112, 456744927232000

Offset: 0

N. J. A. Sloane

nonn

Number of rectangles that can be formed from the vertices of an n-dimensional cube. E.g., a(3)=12 because the three-dimensional cube has six faces plus six rectangles passing through the center of the cube. Cf. A064436: each rectangle on the cube provides an opportunity for a function not to be a linear threshold function, by alternating in value around the rectangle. - Matthew Cook, Jan 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (12, -44, 48).

Third column of triangle A075497.

Cf. A025966.

[6^n/8 - 4^(n-1) + 2^(n-3): n in [0..25]]; // Vincenzo Librandi, Apr 26 2011

[seq(9/2*6^n-4*4^n+1/2*2^n,n=0..20)]; # Detlef Pauly (dettodet(AT)yahoo.de), Dec 04 2001

CoefficientList[Series[x^2/((1 - 2 x) (1 - 4 x) (1 - 6 x)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 31 2018 *)

[((6^n - 2^n)/4-(4^n - 2^n)/2)/2 for n in range(0,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
G.f.: x^2/((1-2*x)*(1-4*x)*(1-6*x)).
E.g.f.: (exp(2*x) - 8*exp(4*x) + 9*exp(6*x))/2!.
a(n) =((6^n - 2^n)/4 - (4^n - 2^n)/2)/2 , n >= 0. - Zerinvary Lajos, Jun 05 2009

A187075 A Galton triangle: T(n,k) = 2kT(n-1,k) + (2k-1)T(n-1,k-1).

Original entry on oeis.org

1, 2, 3, 4, 18, 15, 8, 84, 180, 105, 16, 360, 1500, 2100, 945, 32, 1488, 10800, 27300, 28350, 10395, 64, 6048, 72240, 294000, 529200, 436590, 135135, 128, 24384, 463680, 2857680, 7938000, 11060280, 7567560, 2027025, 256, 97920, 2904000, 26107200, 105099120, 220041360, 249729480, 145945800, 34459425

Offset: 1

Peter Bala, Mar 27 2011

This is a companion triangle to A186695.
Let f(x) = (exp(2*x) + 1)^(-1/2); then the n-th derivative of f equals Sum_{k=1..n} (-1)^k*T(n,k)*(f(x))^(2*k+1). - Groux Roland, May 17 2011
Triangle T(n,k), 1 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013

			Triangle begins
n\k.|...1.....2......3......4......5......6
===========================================
..1.|...1
..2.|...2.....3
..3.|...4....18.....15
..4.|...8....84....180....105
..5.|..16...360...1500...2100....945
..6.|..32..1488..10800..27300..28350..10395
..
Examples of recurrence relation:
T(4,3) = 6*T(3,3) + 5*T(3,2) = 6*15 + 5*18 = 180;
T(6,4) = 8*T(5,4) + 7*T(5,3) = 8*2100 + 7*1500 = 27300.

G. C. Greubel, Table of n, a(n) for the first 50 rows
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv:1204.4963v3 [math.CO]
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

Cf. A075497, A156919, A186695, A211402.

A187075 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 then 2^(n-1); else 2*k*procname(n-1, k) + (2*k-1)*procname(n-1, k-1) ; end if; end proc:seq(seq(A187075(n,k),k = 1..n),n = 1..10);

Flatten[Table[2^(n - 2*k)*Binomial[2 k, k]*k!*StirlingS2[n, k], {n, 10}, {k, 1, n}]] (* G. C. Greubel, Jun 17 2016 *)

# uses[delehamdelta from A084938]
# Adds a first column (1,0,0,0, ...).
def A187075_triangle(n):
    return delehamdelta([(i+1)*int(is_even(i+1)) for i in (0..n)], [i+1 for i in (0..n)])
A187075_triangle(4)  # Peter Luschny, Oct 20 2013

T(n,k) = 2^(n-2*k)*binomial(2k,k)*k!*Stirling2(n,k).
Recurrence relation T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1) with boundary conditions T(1,1) = 1, T(1,k) = 0 for k >= 2.
G.f.: F(x,t) = 1/sqrt((1+x)-x*exp(2*t)) - 1 = Sum_{n >= 1} R(n,x)*t^n/n! = x*t + (2*x+3*x^2)*t^2/2! + (4*x+18*x^2+15*x^3)*t^3/3! + ....
The g.f. F(x,t) satisfies the partial differential equation dF/dt = 2*(x+x^2)*dF/dx + x*F.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = 2*(x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x.
O.g.f. for column k: (2k-1)!!*x^k/Product_{m = 1..k} (1-2*m*x) (compare with A075497). T(n,k) = (2*k-1)!!*A075497(n,k).
The row polynomials R(n,x) = Sum_{k = 1..n} T(n,k)*x^k satisfy R(n,-x-1) = (-1)^n*(1+x)/x*P(n,x) where P(n,x) is the n-th row polynomial of A186695. We also have R(n,x/(1-x)) = (x/(1-x)^n)*Q(n-1,x) where Q(n,x) is the n-th row polynomial of A156919.
T(n,k) = 2^(n-k)*A211608(n,k). - Philippe Deléham, Oct 20 2013

A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 4, 10, 11, 0, 8, 36, 48, 49, 0, 16, 136, 236, 256, 257, 0, 32, 528, 1248, 1508, 1538, 1539, 0, 64, 2080, 6896, 9696, 10256, 10298, 10299, 0, 128, 8256, 39168, 66384, 74784, 75848, 75904, 75905, 0, 256, 32896, 226496, 475136, 586352, 607520, 609368, 609440, 609441

Offset: 0

Mark Wildon, Aug 13 2015

C_t(n) is the number of sequences of t top-to-random shuffles that leave a deck of n cards invariant, if each shuffle is permitted to flip the orientation of the card it moves.
C_t(n) =  where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.
C_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks.
C_t(n) = C_t(t) if n > t.

			Triangle starts:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   10,   11;
  0,  8,   36,   48,   49;
  0, 16,  136,  236,  256,   257;
  0, 32,  528, 1248, 1508,  1538,  1539;
  0, 64, 2080, 6896, 9696, 10256, 10298, 10299;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.

Columns n=0,1,2,3 give A000007, A000079, A007582, A233162 (proved for n=3 in reference above).
Main diagonal gives A004211.
Cf. A075497.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 13 2015

CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];
Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).

C_t(n) = Sum_{i=0..n} A075497(t,i).

A025966 Expansion of 1/((1-2x)(1-4x)(1-6x)(1-8x)).

Original entry on oeis.org

1, 20, 260, 2800, 27216, 248640, 2182720, 18656000, 156544256, 1296655360, 10641146880, 86744985600, 703688298496, 5688011079680, 45855653642240, 368956766617600, 2964331947687936, 23790756829593600

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (20,-140,400,-384).

Fourth column of triangle A075497.

Cf. A016283, A075510.

CoefficientList[Series[1/((1-2x)(1-4x)(1-6x)(1-8x)),{x,0,40}],x] (* Harvey P. Dale, Apr 13 2019 *)

Vec(1/((1-2*x)*(1-4*x)*(1-6*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (2^n)*stirling2(n+4, 4), n>=0, with stirling2(n, m)=A008277(n, m).
a(n) = (-2^n+24*4^n-81*6^n+64*8^n)/3!.
G.f.: 1/((1-2*x)*(1-4*x)*(1-6*x)*(1-8*x)).
E.g.f.: (-exp(2*x)+24*exp(4*x)-81*exp(6*x)+64*exp(8*x))/3!.

A298213 Triangle read by rows, expansion of exp(xexp(z)tan(z)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 12, 32, 12, 1, 0, 41, 160, 110, 20, 1, 0, 142, 856, 900, 280, 30, 1, 0, 685, 4816, 7231, 3360, 595, 42, 1, 0, 3192, 29952, 58632, 37856, 9800, 1120, 56, 1, 0, 19921, 199680, 493100, 416640, 147126, 24192, 1932, 72, 1

Offset: 0

Peter Luschny, Jan 15 2018

			Triangle starts:
0: 1;
1: 0,    1;
2: 0,    2,     1;
3: 0,    5,     6,     1;
4: 0,   12,    32,    12,     1;
5: 0,   41,   160,   110,    20,    1;
6: 0,  142,   856,   900,   280,   30,    1;
7: 0,  685,  4816,  7231,  3360,  595,   42,  1;
8: 0, 3192, 29952, 58632, 37856, 9800, 1120, 56, 1;

T(n,1) = A009739(n), T(n,n) = A002378(n-1).
Row sums are A009248.
Cf. A075497.

gf := exp(x*exp(z)*tan(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A298213_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
seq(A298213_row(n), n=0..8);

cp's OEIS Frontend

A016283 a(n) = 6^n/8 - 4^(n-1) + 2^(n-3).

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A187075 A Galton triangle: T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1).

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A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.

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Comments

Examples

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Programs

Maple

Mathematica

Formula

A025966 Expansion of 1/((1-2x)(1-4x)(1-6x)(1-8x)).

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Author

Keywords

Links

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Programs

Mathematica

PARI

Formula

A298213 Triangle read by rows, expansion of exp(x*exp(z)*tan(z)).

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Author

Keywords

Examples

Crossrefs

Programs

Maple

A187075 A Galton triangle: T(n,k) = 2kT(n-1,k) + (2k-1)T(n-1,k-1).

A298213 Triangle read by rows, expansion of exp(xexp(z)tan(z)).