A111396 -id:A111396 - cp's OEIS frontend

A111373 A generalized Pascal triangle.

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 0, 4, 0, 0, 1, 0, 7, 0, 0, 5, 0, 0, 1, 0, 0, 12, 0, 0, 6, 0, 0, 1, 12, 0, 0, 18, 0, 0, 7, 0, 0, 1, 0, 30, 0, 0, 25, 0, 0, 8, 0, 0, 1, 0, 0, 55, 0, 0, 33, 0, 0, 9, 0, 0, 1, 55, 0, 0, 88, 0, 0, 42, 0, 0, 10, 0, 0, 1, 0, 143, 0, 0, 130, 0, 0, 52, 0, 0, 11, 0, 0, 1

Offset: 0

N. J. A. Sloane, Nov 09 2005

First diagonal is A000012, the all 1's sequence. Second nonzero diagonal is A000027 = n. Third nonzero diagonal is A027379 = n*(n+5)/2 for n>=1, or essentially A000217(n) - 3. Fourth nonzero diagonal is A111396. - Jonathan Vos Post, Nov 10 2005

Row sums are A126042. - Paul Barry, Dec 16 2006

			Triangle begins:
   1;
   0,  1;
   0,  0,  1;
   1,  0,  0,  1;
   0,  2,  0,  0,  1;
   0,  0,  3,  0,  0,  1;
   3,  0,  0,  4,  0,  0,  1;
   0,  7,  0,  0,  5,  0,  0,  1;
   0,  0, 12,  0,  0,  6,  0,  0,  1;
  12,  0,  0, 18,  0,  0,  7,  0,  0,  1;
   0, 30,  0,  0, 25,  0,  0,  8,  0,  0,  1;
   0,  0, 55,  0,  0, 33,  0,  0,  9,  0,  0,  1;
  55,  0,  0, 88,  0,  0, 42,  0,  0, 10,  0,  0,  1;
Production matrix is
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1, 0, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 7, 10.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

First column is A001764. Bears same relation to A001764 as A053121 does to A000108.

Cf. A000012, A000027, A027379, A111396, A126042 (row sums), A167543.

function A111373(n,k)
  if k eq n then return 1;
  elif ((n-k) mod 3) eq 0 then return (3/(n-k))*Binomial(n, Floor((n-k-3)/3))*(k+1);
  else return 0;
  end if; return A111373;
end function;
[A111373(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 30 2022

T[n_, k_]= If[k==n, 1, If[Mod[n-k, 3]==0, (3/(n-k))*Binomial[n, (n-k)/3 -1]*(k + 1), 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 30 2022 *)

def A111373(n,k):
    if(k==n): return 1
    elif ((n-k)%3==0): return (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1)
    else: return 0
flatten([[A111373(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 30 2022

Each term is the sum of the two terms above it to the left and two steps to the right.
From Paul Barry, Dec 16 2006: (Start)
Riordan array (g(x^3),x*g(x^3)) where g(x)=(2/sqrt(3x))*sin(asin(sqrt(27x/4))/3), the g.f. of A001764;
Number triangle T(n,k) = C(3*floor((n+2k)/3)-2k,floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3)-k+ 1)(2*cos(2*pi*(n-k)/3)+1)/3. (End)
Inverse of Riordan array (1/(1+x^3), x/(1+x^3)), A126030. - Paul Barry, Dec 16 2006
G.f. (x*A(x))^k=sum{n>=k, T(n,k)*x^n}, where A(x)=1+x^3*A(x)^3. - Vladimir Kruchinin, Feb 18 2011
From G. C. Greubel, Jul 30 2022: (Start)
T(n, k) = (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1) for ( (n-k) mod 3 ) = 0, otherwise 0, with T(n, n) = 1.
T(n, n-3) = A000027(n-2), n >= 3.
T(n, n-6) = A027379(n-5), n >= 6.
T(n, n-9) = A111396(n-8), n >= 9.
T(n, n-12) = A167543(n+5), n >= 12.
Sum_{k=0..n} T(n, k) = A126042(n). (End)

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Jan 23 2006

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 7, 0, 4, 0, 1, 0, 12, 0, 12, 0, 5, 0, 1, 0, 0, 30, 0, 18, 0, 6, 0, 1, 0, 55, 0, 55, 0, 25, 0, 7, 0, 1, 0, 0, 143, 0, 88, 0, 33, 0, 8, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

			Triangle begins
  1,
  0,  1,
  0,  0,  1,
  0,  1,  0,  1,
  0,  0,  2,  0,  1,
  0,  3,  0,  3,  0,  1,
  0,  0,  7,  0,  4,  0,  1,
  0, 12,  0, 12,  0,  5,  0,  1
From _Paul Barry_, Sep 28 2009: (Start)
Production matrix is
  0, 1,
  0, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001477, A001764, A006013, A055998.

Cf. A047749 (row sums), A098746 (diagonal sums), A124304 (inverse).

A124305:= func< n,k | n eq 0 select 1 else (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial(n + Floor((n-k)/2) -1, n-1) >;
[A124305(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 25 2023

A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]];
Table[A124305[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 19 2023 *)

def A124305(n,k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n
flatten([[A124305(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 25 2023

Sum_{k=0..n} T(n, k) = A047749(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).
From G. C. Greubel, Aug 19 2023: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n-k))*(k/n)*binomial(n + (n-k)/2 - 1, (n-k)/2), with T(0, 0) = 1.
T(n, n) = 1.
T(n, n-2) = A001477(n-2).
T(n, n-4) = A055998(n-4).
T(n, n-6) = A111396(n-6).
T(n, 0) = 0^n.
T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).
T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

Original entry on oeis.org

2, 5, 8, 10, 10, 7, 0, -12, -30, -55, -88, -130, -182, -245, -320, -408, -510, -627, -760, -910, -1078, -1265, -1472, -1700, -1950, -2223, -2520, -2842, -3190, -3565, -3968, -4400, -4862, -5355, -5880, -6438, -7030, -7657, -8320, -9020, -9758, -10535, -11352

Offset: 6

Jamel Ghanouchi, Nov 06 2009

Essentially the same as A111396.

The coefficient of x^(n-6) of the Polynomial B_n(x) defined in A135929.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A135929, A137276.

LinearRecurrence[{4,-6,4,-1},{2,5,8,10},50] (* Harvey P. Dale, May 27 2012 *)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^6*(2 - 3*x)/(x - 1)^4.
a(n) = -A111396(n-12) for n > 11. - Bruno Berselli, Oct 02 2018

Minor edits by N. J. A. Sloane, Nov 09 2009

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A368514 Irregular triangle T(n,k) read by rows: similar to A009766 but length of rows grows like log(3)/log(2).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 0, 1, 3, 3, 1, 4, 7, 0, 1, 5, 12, 12, 0, 1, 6, 18, 30, 30, 1, 7, 25, 55, 85, 0, 1, 8, 33, 88, 173, 173, 1, 9, 42, 130, 303, 476, 0, 1, 10, 52, 182, 485, 961, 961, 0, 1, 11, 63, 245, 730, 1691, 2652, 2652, 1, 12, 75, 320, 1050, 2741, 5393, 8045, 0

Offset: 1

Ruud H.G. van Tol, Dec 28 2023

			Triangle T(n,k) begins:
 n|k:1|  2|  3|  4|  5|  6|  7|  8|...
--+---+---+---+---+---+---+---+---+---
 1|  1
 2|  1   0
 3|  1   1
 4|  1   2   0
 5|  1   3   3
 6|  1   4   7   0
 7|  1   5  12  12   0
 8|  1   6  18  30  30
 9|  1   7  25  55  85   0
10|  1   8  33  88 173 173
11|  1   9  42 130 303 476   0
12|  1  10  52 182 485 961 961   0
...

Ruud H.G. van Tol, Table of n, a(n) for n = 1..11858 (200 rows).
Ruud H.G. van Tol, Sequence on a lattice

Cf. A009766 (Catalan's triangle), A098294 (row lengths), A100982 (row sums).

Cf. A055998, A111396.

row(n) = my(v=Vec([1], logint(3^n, 2)+1-n), c=1); for(i=2, n, for(j=2, c, v[j]+=v[j-1]); c=logint(3^i,2)+1-i); v

rows(n) = my(v=vector(n, i, Vec([1], logint(3^i,2)+1-i))); for(i=3, n, for(j=2, #v[i-1], v[i][j]=v[i][j-1]+v[i-1][j])); v

Row length L(n) = A098294(n) = floor(n*log(3)/log(2)) + 1 - n.
T(n,1) = 1.
T(n+1,k) = T(n+1,k-1) + T(n,k) for 1 < k <= L(n).
T(n+1,L(n+1)) = 0 if L(n+1) > L(n).
T(n+1,2) = n-1.
T(n+3,3) = A055998(n-1) = (n-1)*(n+4)/2.
T(n+5,4) = A111396(n-1) = (n-1)*(n+6)*(n+7)/6.
T(n+1,k) = Sum_{j=1..k} T(n,j) for 1 <= k <= L(n).

Corrected by Ruud H.G. van Tol, Nov 29 2024

A111144 a(n) = n(n+13)(n+14)/6.

Original entry on oeis.org

0, 35, 80, 136, 204, 285, 380, 490, 616, 759, 920, 1100, 1300, 1521, 1764, 2030, 2320, 2635, 2976, 3344, 3740, 4165, 4620, 5106, 5624, 6175, 6760, 7380, 8036, 8729, 9460, 10230, 11040, 11891, 12784, 13720, 14700, 15725, 16796, 17914, 19080, 20295, 21560

Offset: 0

Jonathan Vos Post, Nov 12 2005

After a(1) all values have at least three prime factors with multiplicity, for example a(33) = 11891 = 11 * 23 * 47 and a(49) = 31899 = 3 * 7^3 * 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A111396.

[n*(n+13)*(n+14)/6: n in [0..50]]; // Vincenzo Librandi, Jan 11 2012

Table[n (n + 13) (n + 14)/6, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

a(n)=n*(n+13)*(n+14)/6 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = A000292(n) + 4n^2 + 30n.
G.f.: x*(35 - 60*x + 26*x^2)/(1-x)^4. - Colin Barker, Jan 11 2012
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = 811373/65585520.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/13 - 752153/7287280. (End)

cp's OEIS Frontend

A111373 A generalized Pascal triangle.

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A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).

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A167541 a(n) = -(n - 4)*(n - 5)*(n - 12)/6.

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A368514 Irregular triangle T(n,k) read by rows: similar to A009766 but length of rows grows like log(3)/log(2).

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PARI

PARI

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A111144 a(n) = n*(n+13)*(n+14)/6.

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Magma

Mathematica

PARI

Formula

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

A111144 a(n) = n(n+13)(n+14)/6.