A155856 -id:A155856 - cp's OEIS frontend

A155857 Row sums of triangle A155856.

1, 2, 6, 23, 107, 590, 3786, 27821, 230869, 2137978, 21873854, 245151555, 2987967551, 39358156310, 557259550034, 8440866957273, 136211005966889, 2333068710452146, 42276699542130166, 808068680469402095, 16248405328930779027, 342877404288485770718, 7576652528705018522906

Offset: 0

Paul Barry, Jan 29 2009

For positive n, a(n) equals the permanent of the n X n matrix with 2's along the main diagonal and the upper diagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A155856.

Table[Sum[Binomial[2*n-k,k]*(n-k)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)

[sum(binomial(2*n-k, k)*factorial(n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 05 2021

G.f.: 1/(1 -x -x/(1 -x -x/(1 -x -2*x/(1 -x -2*x/(1 -x -3*x/(1 -x -3*x/(1 - ... (continued fraction);
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*(n-k)!.
a(n) = Sum_{k=0..n} binomial(n+k, 2*k)*k!. - Paul Barry, May 28 2009
a(n) = (n+1)*a(n-1) -(n-3)*a(n-2) -a(n-3). - R. J. Mathar, Nov 15 2012
a(n) ~ exp(2) * n!. - Vaclav Kotesovec, Feb 08 2014

A155858 Diagonal sums of triangle A155856.

Original entry on oeis.org

1, 1, 3, 9, 35, 168, 967, 6538, 50831, 446919, 4383861, 47451921, 561715093, 7217604520, 100031995789, 1487319385140, 23613262336093, 398673670050021, 7132188802005991, 134766129577134553, 2681929390235577831

Offset: 0

Paul Barry, Jan 29 2009

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A155856.

Table[Sum[Binomial[2*n-3*k, k]*(n-2*k)!, {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 05 2021 *)

[sum( binomial(2*n-3*k, k)*factorial(n-2*k) for k in (0..n//2) ) for n in (0..30)] # G. C. Greubel, Jun 05 2021

G.f.: 1/(1 -x^2 -x/(1 -x^2 -x/(1 -x^2 -2*x/(1 -x^2 -2*x/(1 -x^2 -3*x/(1 -x^2 -3*x/(1 - ... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k)*(n-2*k)!.
Conjecture: a(n) -(n-1)*a(n-1) -(n-2)*a(n-2) +(n-3)*a(n-3) +(n-10)*a(n-4) -5*a(n-5) +3*a(n-6) +3*a(n-7) = 0. - R. J. Mathar, Feb 05 2015
a(n) ~ n! * (1 + 2/n + 1/n^2 - 2/(3*n^3) - 22/(3*n^4) - 491/(15*n^5) - 11467/(90*n^6) - ...). - Vaclav Kotesovec, Jun 05 2021

A161556 Exponential Riordan array [1 + (sqrt(Pi)/2)xexp(x^2/4)*erf(x/2), x].

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 6, 0, 30, 0, 15, 0, 1, 0, 42, 0, 70, 0, 21, 0, 1, 24, 0, 168, 0, 140, 0, 28, 0, 1, 0, 216, 0, 504, 0, 252, 0, 36, 0, 1, 120, 0, 1080, 0, 1260, 0, 420, 0, 45, 0, 1

Offset: 0

Paul Barry, Jun 13 2009

Row sums are A084261.

			Triangle begins
   1;
   0,   1;
   1,   0,   1;
   0,   3,   0,   1;
   2,   0,   6,   0,   1;
   0,  10,   0,  10,   0,   1;
   6,   0,  30,   0,  15,   0,   1;
   0,  42,   0,  70,   0,  21,   0,   1;
  24,   0, 168,   0, 140,   0,  28,   0,   1;
Production matrix begins
   0,   1;
   1,   0,   1;
   0,   2,   0,   1;
  -1,   0,   3,   0,   1;
   0,  -4,   0,   4,   0,   1;
   6,   0, -10,   0,   5,   0,   1;
   0,  36,   0, -20,   0,   6,   0,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A155856, A156367.

T[n_, k_] := Boole[k <= n] Binomial[n, k] ((n-k)/2)! (1 + (-1)^(n-k))/2; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 30 2016 *)

T(n,k) = [k<=n]*binomial(n,k)*((n-k)/2)!*(1+(-1)^(n-k))/2.

G.f.: 1/(1-x*y-x^2/(1-x*y-x^2/(1-x*y-2x^2/(1-x*y-2x^2/(1-x*y-3x^2/(1-... (continued fraction).

A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 24, 1, 15, 70, 168, 216, 120, 1, 21, 140, 504, 1080, 1320, 720, 1, 28, 252, 1260, 3960, 7920, 9360, 5040, 1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320, 1, 45, 660, 5544, 30888, 120120, 327600, 604800, 685440, 362880

Offset: 0

Paul Barry, Feb 08 2009

			Triangle begins
  1;
  1,  1;
  1,  3,   2;
  1,  6,  10,    6;
  1, 10,  30,   42,    24;
  1, 15,  70,  168,   216,   120;
  1, 21, 140,  504,  1080,  1320,   720;
  1, 28, 252, 1260,  3960,  7920,  9360,  5040;
  1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320;

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A084261 (diagonal sums), A155856 (row reversal), A155857 (row sums)

Flatten[Table[Binomial[n+k,2k]k!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 17 2015 *)

flatten([[factorial(k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

G.f.: 1/(1 -x -x*y/(1 -x -x*y/(1 -x -2*x*y/(1 -x -2*x*y/(1 -x -3*x*y/(1 -x -3*x*y/(1 - ... (continued fraction).
T(n, k) = binomial(n+k, 2*k)*k!
T(n, k) = A155856(n, n-k).
Sum_{k=0..n} T(n, k) = A155857(n).
sum_{k=0..floor(n/2)} T(n, k) = A084261(n).

A156368 A ménage triangle.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 3, 8, 6, 6, 1, 16, 35, 38, 20, 10, 1, 96, 211, 213, 134, 50, 15, 1, 675, 1459, 1479, 915, 385, 105, 21, 1, 5413, 11584, 11692, 7324, 3130, 952, 196, 28, 1, 48800, 103605, 104364, 65784, 28764, 9090, 2100, 336, 36, 1

Offset: 0

Paul Barry, Feb 08 2009

			Triangle begins:
   1;
   0,   1;
   0,   1,   1;
   1,   1,   3,   1;
   3,   8,   6,   6,  1;
  16,  35,  38,  20, 10,  1;
  96, 211, 213, 134, 50, 15,  1;

A. Kaufmann, Introduction à la combinatorique en vue des applications, p.188-189, Dunod, Paris, 1968. - Philippe Deléham, Apr 04 2014

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

Cf. A000142, A000271, A007318, A155856.

T[n_,k_]:= Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[2*n-j, j]*(n-j)!, {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 05 2021 *)

def A156368(n,k): return sum( (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*factorial(n-j) for j in (0..n) )
flatten([[A156368(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

T(n, k) = Sum_{j=0..n} (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*(n-j)!.
T(n, 0) = A000271(n).
Sum_{k=0..n} T(n, k) = n!.
Equals A155856*A007318^{-1}.
G.f.: 1/(1 +x -x*y -x/(1 +x -x*y -x/(1 +x -x*y -2*x/(1 +x -x*y -2*x/(1 +x -x*y -3*x/(1 +x -x*y -3*x/(1 +x -x*y -4*x/(1 + ... (continued fraction).
G.f.: Sum_{n>=0} n! * x^n/(1 + (1-y)*x)^(2*n+1). - Ira M. Gessel, Jan 15 2013

cp's OEIS Frontend

A155857 Row sums of triangle A155856.

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A155858 Diagonal sums of triangle A155856.

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A161556 Exponential Riordan array [1 + (sqrt(Pi)/2)*x*exp(x^2/4)*erf(x/2), x].

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A156367 Triangle T(n, k) = binomial(n+k, 2*k)*k!, read by rows.

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A156368 A ménage triangle.

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Mathematica

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A161556 Exponential Riordan array [1 + (sqrt(Pi)/2)xexp(x^2/4)*erf(x/2), x].

A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.