A051947 -id:A051947 - cp's OEIS frontend

A050483 Partial sums of A051947.

1, 11, 60, 228, 690, 1782, 4092, 8580, 16731, 30745, 53768, 90168, 145860, 228684, 348840, 519384, 756789, 1081575, 1519012, 2099900, 2861430, 3848130, 5112900, 6718140, 8736975, 11254581, 14369616, 18195760, 22863368, 28521240, 35338512, 43506672, 53241705, 64786371

Offset: 0

Barry E. Williams, Dec 26 1999

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A051947.

Cf. A093561 ((4, 1) Pascal, column m=7).

Table[(4*n + 7)*Binomial[n + 6, 6]/7, {n, 0, 40}] (* Amiram Eldar, Feb 15 2022 *)

a(n) = C(n+6, 6)*(4n+7)/7.
G.f.: (1+3*x)/(1-x)^8. - proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Sum_{n>=0} 1/a(n) = 57344*Pi/663 - 114688*log(2)/221 + 295372/3315. - Amiram Eldar, Feb 15 2022

A093561 (4,1) Pascal triangle.

Original entry on oeis.org

1, 4, 1, 4, 5, 1, 4, 9, 6, 1, 4, 13, 15, 7, 1, 4, 17, 28, 22, 8, 1, 4, 21, 45, 50, 30, 9, 1, 4, 25, 66, 95, 80, 39, 10, 1, 4, 29, 91, 161, 175, 119, 49, 11, 1, 4, 33, 120, 252, 336, 294, 168, 60, 12, 1, 4, 37, 153, 372, 588, 630, 462, 228, 72, 13, 1, 4, 41, 190, 525, 960, 1218

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(4;n,m) gives in the columns m >= 1 the figurate numbers based on A016813, including the hexagonal numbers A000384 (see the W. Lang link).
This is the fourth member, d=4, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 and A093560, for d=1..3.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+3*z)/(1-(1+x)*z).
The SW-NE diagonals give A000285(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018

			Triangle begins
  [1];
  [4, 1];
  [4, 5, 1];
  [4, 9, 6, 1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, >Rows n = 0..125 of triangle, flattened
P. Bala, A note on the diagonals of a proper Riordan Array
W. Lang, First 10 rows and array of figurate numbers .

Cf. Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 3 for n=2 and 0 otherwise.
Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, A051947, A050483, A052181, A055843.
Cf. A007318, A093562 (d=5), A228196, A228576.

a093561 n k = a093561_tabl !! n !! k
a093561_row n = a093561_tabl !! n
a093561_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [4, 1]
-- Reinhard Zumkeller, Aug 31 2014

from math import comb, isqrt
def A093561(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+3*(r-a))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m) = F(4;n-m, m) for 0<= m <= n, otherwise 0, with F(4;0, 0)=1, F(4;n, 0)=4 if n>=1 and F(4;n, m) = (4*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 3*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 9*x + 6*x^2/2! + x^3/3!) = 4 + 13*x + 28*x^2/2! + 50*x^3/3! + 80*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A082680 Triangle read by rows: T(n,k) is the number of 2-stack sortable n-permutations with k runs.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 49, 20, 1, 1, 35, 168, 168, 35, 1, 1, 56, 462, 900, 462, 56, 1, 1, 84, 1092, 3630, 3630, 1092, 84, 1, 1, 120, 2310, 12012, 20449, 12012, 2310, 120, 1, 1, 165, 4488, 34320, 91091, 91091, 34320, 4488, 165, 1, 1, 220, 8151, 87516, 340340, 529984, 340340, 87516, 8151, 220, 1

Offset: 1

Ralf Stephan, May 19 2003

Number of beta(1,0)-trees on n+1 nodes with k leaves.
Row sums are given by A000139. - F. Chapoton, Nov 17 2015
T(n,k) is the number of rooted non-separable planar maps with n+1 edges, k+1 faces and n+2-k vertices. - Andrew Howroyd, Mar 29 2021

			Triangle starts:
  1;
  1,  1;
  1,  4,    1;
  1, 10,   10,    1;
  1, 20,   49,   20,    1;
  1, 35,  168,  168,   35,    1;
  1, 56,  462,  900,  462,   56,  1;
  1, 84, 1092, 3630, 3630, 1092, 84, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
M. Bona, 2-stack sortable permutations with a given number of runs, arXiv:math/9705220 [math.CO], 1997.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
Enrica Duchi, Veronica Guerrini, Simone Rinaldi, and Gilles Schaeffer, Fighting Fish: enumerative properties, arXiv:1611.04625 [math.CO], 2016.

Cf. A000292 (2nd column), A051947 (3rd column).
Cf. A000139 (row sums).
Similar to A008292 and A001263.

Table[(n+k-1)!(2n-k)!/k!/(n+1-k)!/(2k-1)!/(2n-2k+1)!,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jun 10 2020 *)

T(n, k) = (n+k-1)!*(2*n-k)!/k!/(n+1-k)!/(2*k-1)!/(2*n-2*k+1)! \\ Andrew Howroyd, Mar 29 2021

T(n, k) = (n+k-1)!*(2*n-k)!/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!).

Terms a(52) and beyond from Andrew Howroyd, Mar 29 2021

A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

Original entry on oeis.org

1, 6, 1, 15, 7, 1, 28, 22, 8, 1, 45, 50, 30, 9, 1, 66, 95, 80, 39, 10, 1, 91, 161, 175, 119, 49, 11, 1, 120, 252, 336, 294, 168, 60, 12, 1, 153, 372, 588, 630, 462, 228, 72, 13, 1, 190, 525, 960, 1218, 1092, 690, 300, 85, 14, 1, 231, 715, 1485, 2178, 2310, 1782, 990, 385, 99, 15, 1

Offset: 0

Gary W. Adamson, Nov 24 2006

Left border = A000384, hexagonal numbers. The following columns are A002412, A002417, A034263, A051947, ...
Row sums = (1, 7, 23, 59, 135, 291, ...) = A126284.
A125232 is the analogous triangle for the pentagonal numbers.

			First few rows of the triangle:
   1;
   6,   1;
  15,   7,   1;
  28,  22,   8,   1;
  45,  50,  30,   9,  1;
  66,  95,  80,  39, 10,  1;
  91, 161, 175, 119, 49, 11, 1;
  ...
Example: (5,3) = 80 = 30 + 50 = (4,3) + (4,2).

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1964, p. 189.

Cf. A000384, A002412, A002417, A034263, A051947, A125232.

A000384Psum:= proc(n,k) coeftayl( x*(1+3*x)/(1-x)^(3+k),x=0,n) ; end: A125233 := proc(n,k) A000384Psum(n-k,k) ; end: for n from 1 to 15 do for k from 0 to n -1 do printf("%d,",A125233(n,k)) ; od: od: # R. J. Mathar, May 03 2008

T[n_, k_] := T[n, k] = Which[k == 0, n (2 n - 1), 1 <= k < n, T[n - 1, k] + T[n - 1, k - 1], True, 0];
Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Sep 14 2023, after R. J. Mathar *)

T(n,0)=A000384(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>1. - R. J. Mathar, May 03 2008

Edited and extended by R. J. Mathar, May 03 2008, and M. F. Hasler, Sep 29 2012

A112852 Table with row lengths 1 1 2 3 5 9 17 33 65 ... which counts the objects described in A047970 and A112508.

Original entry on oeis.org

1, 2, 4, 1, 7, 6, 1, 11, 20, 8, 3, 1, 16, 50, 33, 21, 10, 3, 6, 4, 1, 22, 105, 98, 81, 49, 21, 48, 36, 12, 3, 6, 12, 10, 4, 10, 5, 1, 29, 196, 238, 231, 168, 81, 210, 168, 68

Offset: 0

Alford Arnold, Sep 24 2005

The row sums are 1 2 5 14 43 144 523 2048 8597 ... A047970. The columns are essentially A000124, A002415, A051836, A112851, A051947, ...

			The table begins
1
2
4 1
7 6 1
11 20 8 3 1
16 50 33 21 10 3 6 4 1
22 105 98 81 49 21

Cf. A000124, A002415, A051836, A112851, A051947, A047970, A112508.

cp's OEIS Frontend

A050483 Partial sums of A051947.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A093561 (4,1) Pascal triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Python

Formula

A082680 Triangle read by rows: T(n,k) is the number of 2-stack sortable n-permutations with k runs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A112852 Table with row lengths 1 1 2 3 5 9 17 33 65 ... which counts the objects described in A047970 and A112508.

Views

Author

Keywords

Comments

Examples

Crossrefs