A000354 -id:A000354 - cp's OEIS frontend

A337552 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (3k-2) a(n-k).

1, 1, 6, 37, 330, 3613, 47652, 732625, 12875118, 254540413, 5591435136, 135108218353, 3561467337546, 101704047315037, 3127751183515020, 103059820083026449, 3622223857996975110, 135266462416766669917, 5348457650664454581240, 223227700948792985989777

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Cf. A000180, A000354, A337553, A337554.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (3 k - 2) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(Exp[x] (2 - 3 x) - 1), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1 / (exp(x + O(x*x^n)) * (2 - 3*x) - 1)))} \\ Andrew Howroyd, Aug 31 2020

E.g.f.: 1 / (exp(x) * (2 - 3*x) - 1).

a(n) ~ n! * c / ((1-c) * (2/3 - c)^(n+1)), where c = -LambertW(-exp(-2/3)/3). - Vaclav Kotesovec, Aug 31 2020

A337553 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4k-3) a(n-k).

Original entry on oeis.org

1, 1, 7, 45, 439, 5157, 73455, 1217101, 23066311, 491680437, 11645898655, 303422639517, 8624098330359, 265546702327813, 8805478883825359, 312844282877905389, 11855836533424581415, 477380986427269453653, 20352680600044759742463, 915923521948522369041469

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Cf. A000354, A001907, A337552, A337554.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (4 k - 3) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(Exp[x] (3 - 4 x) - 2), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1 / (exp(x + O(x*x^n)) * (3 - 4*x) - 2)))} \\ Andrew Howroyd, Aug 31 2020

E.g.f.: 1 / (exp(x) * (3 - 4*x) - 2).

a(n) ~ n! * c * 2^(2*n+1) / ((1-c) * (3 - 4*c)^(n+1)), where c = -LambertW(-exp(-3/4)/2). - Vaclav Kotesovec, Aug 31 2020

A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5k-4) a(n-k).

Original entry on oeis.org

1, 1, 8, 53, 560, 6961, 105898, 1867393, 37713620, 856269401, 21606253238, 599664843433, 18156702186880, 595557844417441, 21037627605306578, 796218790808110673, 32143778726932363340, 1378765268603813275081, 62619174356163136219918, 3001963660666272082265113

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Cf. A000354, A001908, A337552, A337553.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (5 k - 4) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(Exp[x] (4 - 5 x) - 3), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1 / (exp(x + O(x*x^n)) * (4 - 5*x) - 3)))} \\ Andrew Howroyd, Aug 31 2020

E.g.f.: 1 / (exp(x) * (4 - 5*x) - 3).

a(n) ~ n! * c / (3*(1-c) * (4/5 - c)^(n+1)), where c = -LambertW(-3*exp(-4/5)/5). - Vaclav Kotesovec, Aug 31 2020

A090628 Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 6, 1, 1, 1, 10, 29, 24, 1, 1, 1, 17, 82, 233, 120, 1, 1, 1, 26, 177, 1000, 2329, 720, 1, 1, 1, 37, 326, 2913, 14968, 27949, 5040, 1, 1, 1, 50, 541, 6776, 58017, 269488, 391285, 40320, 1, 1, 1, 65, 834, 13609, 168376, 1393137, 5659120, 6260561, 362880, 1

Offset: 0

Philippe Deléham, Dec 13 2003

			Row n=0: 1,  1,   1,    1,    1,    1,     1,     1, ...
Row n=1: 1,  1,   1,    1,    1,    1,     1,     1, ...
Row n=2: 1,  2,   5,   10,   17,   26,    37,    50, ...
Row n=3: 1,  6,  29,   82,  177,  326,   541,   834, ...
Row n=4: 1, 24, 233, 1000, 2913, 6776, 13609, 24648, ...

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

Cf. A008290.

Columns: A000012, A000142, A000354.

T:= (n, k)-> `if`(n=0, 1, LinearAlgebra[Permanent](
              Matrix(n, (i, j)-> `if`(i=j, 1, k)))):
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jul 09 2017
# second Maple program:
b:= proc(n, k) b(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (b(n-1, 0)+b(n-2, 0))), binomial(n, k)*b(n-k, 0))
    end:
T:= proc(n, k) T(n, k):= add(b(n, j)*k^(n-j), j=0..n) end:
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jul 09 2017

T[0, _] = 1;
T[n_, k_] := Permanent[Table[If[i == j, 1, k], {i, n}, {j, n}]];
Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

T(n,k) = matpermanent(matrix(n, n, i, j, if (i==j, 1, k)));
matrix(10, 10, n, k, T(n,k)) \\ Michel Marcus, Dec 07 2019

T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j).

3 terms corrected and more terms from Alois P. Heinz, Jul 09 2017

A097821 Expansion of e.g.f. exp(2x)/(1-5x).

Original entry on oeis.org

1, 7, 74, 1118, 22376, 559432, 16783024, 587405968, 23496238976, 1057330754432, 52866537722624, 2907659574746368, 174459574484786176, 11339872341511109632, 793791063905777690624, 59534329792933326829568

Offset: 0

Paul Barry, Aug 26 2004

Second binomial transform of n!*5^n.

Robert Israel, Table of n, a(n) for n = 0..353

Cf. A056545, A000354.

f:= proc(n) option remember; 5*n*procname(n-1)+2^n end proc:
f(0):= 1:
map(f, [$0..50]); # Robert Israel, Nov 10 2022

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-5*x))) \\ Michel Marcus, Nov 08 2022

a(n) = 5*n*a(n-1) + 2^n, n > 0, a(0)=1.
D-finite with recurrence a(n) +(-5*n-2)*a(n-1) +10*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) = 5^n * n! * Sum_{k = 0..n} (2/5)^k/k! = 5^n * exp(2/5) * gamma(n + 1, 2/5). - Gerry Martens, Nov 07 2022

A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 29, 76, 233, 685, 2329, 7534, 27949, 97943, 391285, 1469144, 6260561, 24975449, 112690097, 474533530, 2253801941, 9965204131, 49583642701, 229199695012, 1190007424825, 5729992375301, 30940193045449, 154709794133126, 866325405272573

Offset: 0

Ryan Brooks, May 06 2020

nonn

			a(5) = (5*3*1)*(1/(1) - 1/(3*1) + 1/(5*3*1)) = 15-5+1 = 11.

Alois P. Heinz, Table of n, a(n) for n = 0..807

Even bisection gives A000354.

Cf. A000166, A006882.

a:= proc(n) option remember; `if`(n<2, [0$2, 1$2][n+3],
      (n-1)*a(n-2)+(n-2)*a(n-4))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, May 06 2020

RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n a[n-2] + (-1)^Floor[n/2]}, a, {n, 0, 32}] (* Jean-François Alcover, Nov 27 2020 *)

a(n) = n*a(n-2) + (-1)^floor(n/2).
a(2n) = A000354(n).
From Ryan Brooks, Oct 25 2020: (Start)
a(2n)/A006882(2n) ~ 1/sqrt(e) = A092605.
a(2n+1)/A006882(2n+1) ~ sqrt(Pi/(2*e))*erfi(1/sqrt(2)) = A306858. (End)

A342381 Triangle read by rows: T(n,k) is the number of symmetries of the n-dimensional hypercube that fix exactly 2*k facets; n,k >= 0.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 29, 15, 3, 1, 233, 116, 30, 4, 1, 2329, 1165, 290, 50, 5, 1, 27949, 13974, 3495, 580, 75, 6, 1, 391285, 195643, 48909, 8155, 1015, 105, 7, 1, 6260561, 3130280, 782572, 130424, 16310, 1624, 140, 8, 1, 112690097, 56345049, 14086260, 2347716, 293454, 29358, 2436, 180, 9, 1

Offset: 0

Peter Kagey, Mar 09 2021

Equivalently the number of symmetries of the n-dimensional cross-polytope that fix exactly 2*k vertices.

If a facet of the hypercube is fixed, then the opposite facet must also be fixed.

			Table begins:
n\k |         0        1        2       3      4     5    6   7 8 9
----+--------------------------------------------------------------
  0 |         1
  1 |         1        1
  2 |         5        2        1
  3 |        29       15        3       1
  4 |       233      116       30       4      1
  5 |      2329     1165      290      50      5     1
  6 |     27949    13974     3495     580     75     6    1
  7 |    391285   195643    48909    8155   1015   105    7   1
  8 |   6260561  3130280   782572  130424  16310  1624  140   8 1
  9 | 112690097 56345049 14086260 2347716 293454 29358 2436 180 9 1
For the cube in n=2 dimensions (the square) there is
T(2,2) = 1 symmetry that fixes all 2*2 = 4 sides, namely the identity:
     2
   +---+
  3|   |1;
   +---+
     4
T(2,1) = 2 symmetries that fix 2*1 = 2 sides, namely horizonal/vertical flips:
     4           2
   +---+       +---+
  3|   |1 and 1|   |3;
   +---+       +---+
     2           4
and T(2,0) = 5 symmetries that fix 2*0 = 0 sides, namely rotations or diagonal flips:
     1         4         3         3            1
   +---+     +---+     +---+     +---+        +---+
  2|   |4,  1|   |3,  4|   |2,  2|   |4, and 4|   |2.
   +---+     +---+     +---+     +---+        +---+
     3         2         1         1            3

Peter Kagey, Rows n = 0..100, flattened
Wikipedia, Cross-polytope
Wikipedia, Hypercube
Wikipedia, Hyperoctahedral group

Columns and diagonals: A000354 (k=0), A161937 (k=1), A028895 (n=k+2).
Row sums are A000165.
Cf. A001147, A114320.

f(n) = sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k); \\ A000354
T(n, k) = f(n-k)*binomial(n, k); \\ Michel Marcus, Mar 10 2021

T(n,k) = A114320(2n,k)/A001147(n).

T(n,k) = A000354(n-k)*binomial(n,k).

A343582 a(n) = (-1)^nn![x^n] exp(-3x)/(1 - 2x).

Original entry on oeis.org

1, 1, 5, -3, 105, -807, 10413, -143595, 2304081, -41453775, 829134549, -18240782931, 437779321785, -11382260772087, 318703306401405, -9561099177693243, 305955173729230497, -10402475906664696735, 374489132640316502949, -14230587040330864850595, 569223481613238080808201

Offset: 0

Peter Luschny, Apr 24 2021

sign

The row polynomials of the rencontres numbers (A008290) evaluated at -1/2 and normalized by (-2)^n.

Cf. A008290, A000166, A000354.

egf := exp(-3*x)/(1 - 2*x): ser := series(egf, x, 32):
seq((-1)^n*n!*coeff(ser, x, n), n=0..20);

a[n_] := (-2)^n Sum[Binomial[n, k] Subfactorial[n - k] (-2)^(-k), {k, 0, n}];
Table[a[n], {n, 0, 20}]

def A343582():
    a, b, n = 1, 5, 3
    yield 1
    yield a
    while True:
        yield b
        a, b = b, 6*(n - 1)*a - (2*n - 3)*b
        n += 1
a = A343582(); print([next(a) for _ in range(21)])

a(n) = (-2)^n*Sum_{k=0..n} binomial(n, k)*subfactorial(n - k)*(-1/2)^k.

a(n) = 6*(n - 1)*a(n - 2) - (2*n - 3)*a(n - 1) for n >= 3.

A372723 Triangle read by rows: Column k has e.g.f. t^k / ((1 - t)^(k + 1) * exp(t)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 2, 9, 12, 6, 9, 44, 84, 72, 24, 44, 265, 640, 780, 480, 120, 265, 1854, 5430, 8520, 7560, 3600, 720, 1854, 14833, 50988, 97650, 112560, 78120, 30240, 5040, 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320

Offset: 0

Peter Luschny, May 21 2024

			Triangle starts:
[0]     1;
[1]     0,      1;
[2]     1,      2,      2;
[3]     2,      9,     12,       6;
[4]     9,     44,     84,      72,      24;
[5]    44,    265,    640,     780,     480,     120;
[6]   265,   1854,   5430,    8520,    7560,    3600,    720;
[7]  1854,  14833,  50988,   97650,  112560,   78120,  30240,   5040;
[8] 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320;

Cf. A000166 (column 0), A000142 (main diagonal), A062119 (subdiagonal), A000354 (row sums), A033999 (alternating row sums), A372716 (central terms).

MAX := 14; gf := k -> t^k / ((1 - t)^(k + 1) * exp(t)):
ser := k -> series(gf(k), t, MAX):
col := k -> local n; seq(n!*coeff(series(ser(k), t, MAX-1), t, n), n = 0..MAX-2):
T := (n, k) -> col(k)[n+1]:
seq(lprint(seq(T(n, k), k = 0..n)), n = 0..8);

A064670 Triangle T(n,k) (1 <= k <= n) where the first column (T(n,1)) is the sequence of secant numbers A000364.

Original entry on oeis.org

1, 1, 4, 5, 20, 36, 61, 244, 504, 576, 1385, 5540, 11916, 17280, 14400, 50521, 202084, 442224, 697536, 792000, 518400, 2702765, 10811060, 23870196, 39202560, 50198400, 47174400, 25401600, 199360981, 797443924, 1769923944

Offset: 1

Jose L. Arregui (arregui(AT)posta.unizar.es), Oct 09 2001

Replacing m*m by m*(m+1) in the formula, the first column gives the tangent numbers A000182.

			T(4,3) = 9*(T(3,2) + T(3,3)) = 9*(20+36) = 504.

Cf. A064190, A000182, A000354.

T[1, 1] = 1; T[n_, k_] /; 1 <= k <= n := T[n, k] = k^2*Sum[T[n-1, j], {j, k-1, n-1}]; T[, ] = 0; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 25 2016 *)

T(n+1, m) = m*m*Sum_{k = m-1..n} T(n, k) (T(n, 0) = 0).

More terms from Philippe Deléham, Sep 22 2005

cp's OEIS Frontend

A337552 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (3*k-2) * a(n-k).

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A337553 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4*k-3) * a(n-k).

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A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5*k-4) * a(n-k).

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A090628 Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.

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A097821 Expansion of e.g.f. exp(2x)/(1-5x).

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A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.

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A342381 Triangle read by rows: T(n,k) is the number of symmetries of the n-dimensional hypercube that fix exactly 2*k facets; n,k >= 0.

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A343582 a(n) = (-1)^n*n!*[x^n] exp(-3*x)/(1 - 2*x).

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A337552 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (3k-2) a(n-k).

A337553 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4k-3) a(n-k).

A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5k-4) a(n-k).

A343582 a(n) = (-1)^nn![x^n] exp(-3x)/(1 - 2x).