A118931 -id:A118931 - cp's OEIS frontend

A118932 E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).

1, 1, 1, 3, 9, 21, 81, 351, 1233, 10249, 75841, 388411, 3733401, 33702813, 215375889, 1984583511, 19181083041, 141963117201, 1797976123393, 22534941675379, 202605151063081, 2992764505338021, 43182110678814801, 445326641624332623

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals invariant column vector V that satisfies matrix product A118931*V = V, where A118931(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0; thus a(n) = Sum_{k=0..floor(n/3)} A118931(n,k)*a(k), with a(0) = 1.

			E.g.f. A(x) = exp( x + x^3/3 + x^9/3^4 + x^27/3^13 + x^81/3^40 + ...)
= 1 + 1*x + 1*x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 81*x^6/6! + ...

G. C. Greubel, Table of n, a(n) for n = 0..490

Cf. A118931.

Variants: A118930, A118935.

function a(n)
     F:=Factorial;
      if n eq 0 then return 1;
    else return (&+[F(n)*a(j)/(3^j*F(j)*F(n-3*j)): j in [0..Floor(n/3)]]);
      end if; return a; end function;
[a(n): n in [0..25]]; // G. C. Greubel, Mar 07 2021

a[n_]:= a[n]= If[n==0, 1, Sum[n!*a[k]/(3^k*k!*(n-3*k)!), {k, 0, Floor[n/3]}] ];
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Mar 07 2021 *)

{a(n) = if(n==0,1,sum(k=0,n\3,n!/(k!*(n-3*k)!*3^k)*a(k)))}
for(n=0,30,print1(a(n),", "))

/* Defined by E.G.F.: */
{a(n) = n!*polcoeff( exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k)/3^((3^k-1)/2))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))

@CachedFunction
def a(n):
    f=factorial;
    if n==0: return 1
    else: return sum( f(n)*a(k)/(3^k*f(k)*f(n-3*k)) for k in (0..n/3))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021

a(n) = Sum_{k=0..floor(n/3)} (n!/(k!*(n-3*k)!*3^k)) * a(k), with a(0)=1.

A118933 Triangle, read by rows, where T(n,k) = n!/(k!(n-4k)!4^k) for n>=4k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 30, 1, 90, 1, 210, 1, 420, 1260, 1, 756, 11340, 1, 1260, 56700, 1, 1980, 207900, 1, 2970, 623700, 1247400, 1, 4290, 1621620, 16216200, 1, 6006, 3783780, 113513400, 1, 8190, 8108100, 567567000, 1, 10920, 16216200, 2270268000, 3405402000

Offset: 0

Paul D. Hanna, May 06 2006

Row n contains 1+floor(n/4) terms. Row sums yield A118934. Given column vector V = A118935, then V is invariant under matrix product T*V = V, or, A118935(n) = Sum_{k=0..n} T(n,k)*A118935(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(4n,n) = (4*n)!/(n!*4^n), 0 otherwise (cf. A100861 formula due to Paul Barry).

			Triangle begins:
  1;
  1;
  1;
  1;
  1,    6;
  1,   30;
  1,   90;
  1,  210;
  1,  420,   1260;
  1,  756,  11340;
  1, 1260,  56700;
  1, 1980, 207900;
  1, 2970, 623700, 1247400; ...

G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Cf. A118934 (row sums), A118935 (invariant vector).

Variants: A100861, A118931.

F:= Factorial;
[n lt 4*k select 0 else F(n)/(4^k*F(k)*F(n-4*k)): k in [0..Floor(n/4)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

T[n_, k_]:= If[n<4*k, 0, n!/(4^k*k!*(n-4*k)!)];
Table[T[n, k], {n,0,20}, {k,0,n/4}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

T(n,k)=if(n<4*k,0,n!/(k!*(n-4*k)!*4^k))

f=factorial;
flatten([[0 if n<4*k else f(n)/(4^k*f(k)*f(n-4*k)) for k in [0..n/4]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

E.g.f.: A(x,y) = exp(x + y*x^4/4).

A118394 Triangle T(n,k) = n!/(k!(n-3k)!), for n >= 3*k >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400

Offset: 0

Paul D. Hanna, May 07 2006

Row sums form A118395.

Eigenvector is A118396.

			Triangle begins:
  1;
  1;
  1;
  1,    6;
  1,   24;
  1,   60;
  1,  120,    360;
  1,  210,   2520;
  1,  336,  10080;
  1,  504,  30240,    60480;
  1,  720,  75600,   604800;
  1,  990, 166320,  3326400;
  1, 1320, 332640, 13305600, 19958400;
  ...

G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Cf. A118395 (row sums), A118396 (eigenvector).

Variants: A059344, A118931.

F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)

T(n,k)=if(n<3*k || k<0,0,n!/k!/(n-3*k)!)

f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

E.g.f.: A(x,y) = exp(x + y*x^3).

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 8, 8, 1, 5, 10, 10, 5, 1, 20, 40, 20, 1, 6, 15, 20, 15, 6, 1, 40, 120, 120, 40, 40, 1, 7, 21, 35, 35, 21, 7, 1, 70, 280, 420, 280, 70, 280, 280, 1, 8, 28, 56, 70, 56, 28, 8, 1, 112, 560, 1120, 1120, 560, 112, 1120, 2240, 1120

Offset: 0

Peter Luschny, Jun 04 2021

Consider a sequence of Pascal tetrahedrons (depending on a parameter m >= 1), where the slices of the pyramid are scaled. They are given by the e.g.f.s exp(t^m / m) * exp(t*(x + y)), which provide a sequence of bivariate polynomials in x and y, whose monomials are to be ordered in degree-lexicographic order. For m = 1 one gets A109649 (resp. A046816), for m = 2 one gets A344911 (resp. A344678), and for m = 3 the current triangle. The row sums have an unexpected interpretation in A336614 (see the link).

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  3,  1,  2;
[4] 1, 4,  6,  4,  1,  8,  8;
[5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
[6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
[7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
.
p_{6}(x, y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 + 40*x^3 + 120*x^2*y + 120*x*y^2 + 40*y^3 + 40.

mjqxxxx, Proof of conjectured formulas for A336614, Mathematics Stack Exchange.

m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].
m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].
m=3: this triangle      [row sums A336614], scaling A118931 [row sums A001470].

B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);

gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
P[n_] := Expand[n! Coefficient[ser, t, n]];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten

cp's OEIS Frontend

A118932 E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).

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A118933 Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.

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A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

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A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3*k} (n! * binomial(n - 3*k, j)) / (k!*(n - 3*k)!*3^k).

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A118933 Triangle, read by rows, where T(n,k) = n!/(k!(n-4k)!4^k) for n>=4k>=0.

A118394 Triangle T(n,k) = n!/(k!(n-3k)!), for n >= 3*k >= 0, read by rows.

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).