A062748 -id:A062748 - cp's OEIS frontend

A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0

Offset: 0

N. J. A. Sloane

Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012

			Triangle begins:
   0;
   0, 0;
   0, 1,  0;
   0, 2,  2,  0;
   0, 3,  5,  3,  0;
   0, 4,  9,  9,  4,  0;
   0, 5, 14, 19, 14,  5, 0;
   0, 6, 20, 34, 34, 20, 6, 0;
   ...
Seen as a square array read by antidiagonals:
  [0] 0, 0,  0,  0,   0,   0,   0,    0,    0,    0,    0,     0, ... A000004
  [1] 0, 1,  2,  3,   4,   5,   6,    7,    8,    9,   10,    11, ... A001477
  [2] 0, 2,  5,  9,  14,  20,  27,   35,   44,   54,   65,    77, ... A000096
  [3] 0, 3,  9, 19,  34,  55,  83,  119,  164,  219,  285,   363, ... A062748
  [4] 0, 4, 14, 34,  69, 125, 209,  329,  494,  714, 1000,  1364, ... A063258
  [5] 0, 5, 20, 55, 125, 251, 461,  791, 1286, 2001, 3002,  4367, ... A062988
  [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Milan Janjic, Two Enumerative Functions

Triangle without zeros: A014430.
Related: A323211 (A007318(n, k) + 1).
A000295 (row sums), A059841 (alternating row sums), A030662(n-1) (central terms).
Columns include A000096, A062748, A062988, A063258.
Diagonals of A(n, n+d): A030662 (d=0), A010763 (d=1), A322938 (d=2).
Cf. A000004, A001477, A007318, A030662, A059841, A109128, A124089, A129696.

a014473 n k = a014473_tabl !! n !! k
a014473_row n = a014473_tabl !! n
a014473_tabl = map (map (subtract 1)) a007318_tabl
-- Reinhard Zumkeller, Apr 10 2012

[Binomial(n,k)-1: k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008
# The rows of the square array:
Arow := proc(n, len) local gf, ser;
gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019

Table[Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

flatten([[binomial(n,k)-1 for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005
T(n,k) = A109128(n,k) - A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015
If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019
From G. C. Greubel, Apr 08 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n-k, k) = A129696(n-2).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n-1), where b(n) is the repeating pattern {0, 0, -1, -2, -1, 1, 1, -1, -2, -1, 0, 0}_{n=0..11}, with b(n) = b(n-12). (End)

More terms from Erich Friedman

A063258 a(n) = binomial(n+5,4) - 1.

Original entry on oeis.org

4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, 1819, 2379, 3059, 3875, 4844, 5984, 7314, 8854, 10625, 12649, 14949, 17549, 20474, 23750, 27404, 31464, 35959, 40919, 46375, 52359, 58904, 66044, 73814, 82250, 91389, 101269, 111929, 123409, 135750

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 4}_{3}, n >= 0.
If X is an n-set and Y a fixed (n-4)-subset of X then a(n-5) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
For n>=5, a(n-5) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...01000 (the first n-5 zeros), or, the same, a(n-5) is up-down coefficient {n,8} (see comment in A060351). - Vladimir Shevelev, Feb 18 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Guillaume Aupy, Julien Herrmann. Periodicity in optimal hierarchical checkpointing schemes for adjoint computations. Optimization Methods and Software, Volume 32, 2017 - Issue 3. Preprint
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Fifth column (r=4) of FS(4) staircase array A062750.
A column of triangle A014473.
Cf. A000096, A062748, A062751.

[Binomial(n+5,4) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024

[seq(binomial(n+5,4)-1,n=0..37)]; # Zerinvary Lajos, Nov 25 2006

Binomial[5+Range[0,50],4] -1 (* G. C. Greubel, Apr 22 2024 *)

{ for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ Harry J. Smith, Aug 19 2009

[binomial(n+5,4) -1 for n in range(51)] # G. C. Greubel, Apr 22 2024

a(n) = A062750(n+2, 4) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4!.
G.f.: (2-x)*(2-2*x+x^2)/(1-x)^5 = N(4;1, x)/(1-x)^5 with N(4;1, x)= 4 - 6*x + 4*x^2 - x^3, polynomial of second row of A062751.
E.g.f.: (1/24)*(96 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). - G. C. Greubel, Apr 22 2024
a(n) = A000332(n+5)-1. - R. J. Mathar, Nov 22 2024

Simpler definition from Vladeta Jovovic, Jul 21 2003

A062745 Generalized Catalan array FS(3; n,r).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 6, 9, 12, 12, 12, 1, 4, 10, 19, 31, 43, 55, 55, 55, 1, 5, 15, 34, 65, 108, 163, 218, 273, 273, 273, 1, 6, 21, 55, 120, 228, 391, 609, 882, 1155, 1428, 1428, 1428, 1, 7, 28, 83, 203, 431, 822, 1431, 2313, 3468, 4896, 6324, 7752, 7752

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this array appears in Table 2, p. 143 and is called {n over r}_{m-1}, with m=3.
The step width sequence of this staircase array is [1,2,2,2,....], i.e., the degree of the row polynomials is [0,2,4,6,...] = A005843.
The columns r=0..5 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A062748, A005718, A062749.
Number of lattice paths from (0,0) to (r,n) using steps h=(1,0), v=(0,1) and staying on or above the line y = x/2. Example: a(3,2)=6 because from (0,0) to (2,3) we have the following valid paths: vvvhh, vvhvh, vvhhv, vhvvh, vhvhvh and vhvvh (see the Niederhausen reference). - Emeric Deutsch, Jun 24 2005

			Array begins:
  {1};
  {1,1,1};
  {1,2,3,3,3};
  {1,3,6,9,12,12,12};
  ...;
N(3; 1,x) = 3-3*x+x^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Wolfdieter Lang, First 10 rows.
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table 2).
Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.

Cf. A009766, A280759 (rows reversed).
Cf. A062746, A062747, A062748, A062749.
Cf. A005843, A000012, A000027, A000217, A005718.

a:=proc(n,r) if r<=2*n then binomial(n+r,r)-(-1)^(r-1)*sum(binomial(3*i,i)*binomial(i-n-1,r-1-2*i)/(2*i+1),i=0..floor((r-1)/2)) else 0 fi end: for n from 0 to 8 do seq(a(n,r),r=0..2*n) od; # yields sequence in triangular form # Emeric Deutsch, Jun 24 2005

a[0, 0] = 1; a[, -1] = 0; a[n, r_] /; r > 2*n = 0; a[n_, r_] := a[n, r] = a[n, r-1] + a[n-1, r]; Table[a[n, r], {n, 0, 7}, {r, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

a(0,0)=1, a(n,-1)=0, n >= 1; a(n,r) = a(n, r-1) + a(n-1, r) if r <= 2n, 0 otherwise.
G.f. for column r = 2*k+j, k >= 0, j=1, 2: (x^(k+1))*N(3; k, x)/ (1-x)^(2*k+1+j), with row polynomials N(3; k, x) of array A062746; for column r=0: 1/(1-x).
a(n,r) = binomial(n+r, r) - (-1)^(r-1)*Sum_{i=0..floor((r-1)/2)} binomial(3i, i)*binomial(i-n-1, r-1-2i)/(2i+1), 0 <= r <= 2n (see the Niederhausen reference, eq. (17)). - Emeric Deutsch, Jun 24 2005

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A220032 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or diagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..1 nXk array.

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 4, 6, 6, 5, 5, 9, 10, 8, 6, 6, 12, 19, 15, 10, 7, 7, 15, 30, 34, 21, 12, 8, 8, 18, 42, 61, 55, 28, 14, 9, 9, 21, 55, 95, 111, 83, 36, 16, 10, 10, 24, 69, 137, 192, 187, 119, 45, 18, 11, 11, 27, 84, 187, 302, 358, 297, 164, 55, 20, 12, 12, 30, 100, 246, 442, 613, 626

Offset: 1

R. H. Hardin Dec 03 2012

Table starts
..2..2..3...4....5....6.....7.....8.....9....10....11....12....13....14...15
..3..4..6...9...12...15....18....21....24....27....30....33....36....39...42
..4..6.10..19...30...42....55....69....84...100...117...135...154...174..195
..5..8.15..34...61...95...137...187...246...315...395...487...592...711..845
..6.10.21..55..111..192...302...442...618...838..1111..1447..1857..2353.2948
..7.12.28..83..187..358...613...962..1426..2034..2823..3839..5137..6782.8850
..8.14.36.119..297..626..1165..1963..3088..4630..6711..9492.13175.18010
..9.16.45.164..450.1038..2094..3789..6334..9995.15133.22239.31956
.10.18.55.219..656.1646..3587..6962.12375.20581.32588
.11.20.66.285..926.2513..5893.12243.23132.40583
.12.22.78.363.1272.3714..9335.20705.41537
.13.24.91.454.1707.5337.14323.33819

			Some solutions for n=3 k=4
..0..0..0..0....1..0..0..0....1..0..0..0....0..0..0..0....1..0..0..0
..0..0..0..0....1..1..0..0....1..0..0..0....1..1..0..0....1..1..1..0
..1..1..0..0....1..1..0..0....1..0..0..0....1..1..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..198

Column 1 is A000027(n+1)
Column 2 is A004275(n+1)
Column 3 is A000217(n+1)
Column 4 is A062748 for n>1
Row 1 is A000027
Row 2 is A204502(n+3)

A014430 Subtract 1 from Pascal's triangle, read by rows.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 14, 19, 14, 5, 6, 20, 34, 34, 20, 6, 7, 27, 55, 69, 55, 27, 7, 8, 35, 83, 125, 125, 83, 35, 8, 9, 44, 119, 209, 251, 209, 119, 44, 9, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 11, 65, 219, 494, 791, 923, 791, 494, 219, 65, 11

Offset: 0

N. J. A. Sloane

Each value of the sequence (T(x,y)) is equal to the sum of all values in Pascal's Triangle that are in the rectangle defined by the tip (0,0) and the position (x,y). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
To clarify T(n,k) and A129696: We subtract I = Identity matrix from Pascal's triangle to obtain the beheaded variant, A074909. Then take column sums starting from the top of A074909 to get triangle A014430. Row sums of the inverse of triangle T(n,k) gives the Bernoulli numbers, A027641/A026642. Alternatively, triangle T(n,k) as an infinite lower triangular matrix * [the Bernoulli numbers as a vector] = [1, 1, 1, ...]. Given the B_n version starting (1, 1/2, 1/6, ...) triangle T(n,k) * the B_n vector [1, 1/2, 1/6, 0, -1/30, ...] = the triangular numbers. - Gary W. Adamson, Mar 13 2012
From R. J. Mathar, Apr 25 2016: (Start)
If regarded as a symmetric array of the form
  1  2   3   4    5  ...
  2  5   9  14   20  ...
  3  9  19  34   55  ...
  4 14  34  69  125  ...
  5 20  55 125  251  ...
  6 27  83 209  461  ...
  7 35 119 329  791  ...
  8 44 164 494 1286  ...
  9 54 219 714 2001  ...
it contains the rows (and columns) A000096, A062748, A063258, A062988, A124089, ..., A035927 and so on and counts the multisets of digits of numbers in base b>=2 with d>=1 digits (equivalent to the comment in A035927). (End)
Proof of Florian Kleedorfer's formula:  Take sums of the columns of the rectangle - these are all binomial coefficients by the Hockey Stick Identity. Note the locations of these coefficients:  They form a row going almost all the way to the edge, only missing the 1 - apply the Hockey Stick Identity again. - James East, Jul 03 2020

			Triangle begins:
  1;
  2,  2;
  3,  5,  3;
  4,  9,  9,   4;
  5, 14, 19,  14,   5;
  6, 20, 34,  34,  20,  6;
  7, 27, 55,  69,  55, 27,  7;
  8, 35, 83, 125, 125, 83, 35, 8;

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Cf. A000096, A007318, A014430, A026642, A027641, A027642, A035927.
Cf. A062748, A063258, A062988, A074909, A124089, A124326, A129696.
Triangle with zeros: A014473.
Cf. A000295 (row sums).

a014430 n k = a014430_tabl !! n !! k
a014430_row n = a014430_tabl !! n
a014430_tabl = map (init . tail) $ drop 2 a014473_tabl
-- Reinhard Zumkeller, Apr 10 2012

[Binomial(n+2,k+1)-1: k in [0..n], n in [0..13]]; // G. C. Greubel, Feb 25 2023

Table[Sum[Sum[Binomial[m, j], {m, j, j+(n-k)}], {j,0,k}], {n,0,10}, {k, 0,n}]//Flatten (* Michael De Vlieger, Sep 01 2020 *)
Table[Binomial[n+2,k+1] -1, {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 25 2023 *)

flatten([[binomial(n+2,k+1)-1 for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Feb 25 2023

T(n, k) = T(n-1, k) + T(n-1, k-1) + 1, T(0, 0)=1. - Ralf Stephan, Jan 23 2005
G.f.: 1 / ((1-x)*(1-x*y)*(1-x*(1+y))). - Ralf Stephan, Jan 24 2005
T(n, k) = Sum_{j=0..k} Sum_{m=j..j+(n-k)} binomial(m, j). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
T(n, k) = binomial(n+2, k+1) - 1. - G. C. Greubel, Feb 25 2023

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Apr 10 2012

A062988 a(n) = binomial(n+6,5) - 1.

Original entry on oeis.org

5, 20, 55, 125, 251, 461, 791, 1286, 2001, 3002, 4367, 6187, 8567, 11627, 15503, 20348, 26333, 33648, 42503, 53129, 65779, 80729, 98279, 118754, 142505, 169910, 201375, 237335, 278255, 324631

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 5}_{4}, n >= 0.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Sixth column (r=5) of FS(5) staircase array A062985.
A column of triangle A014473.
Cf. A000096, A062748, A063258.

[Binomial(n+6,5) -1: n in [0..40]]; // G. C. Greubel, Apr 25 2024

[seq(binomial(n+6,5)-1, n=0..35)]; # Zerinvary Lajos, Nov 25 2006

Binomial[Range[6,45],5] -1 (* G. C. Greubel, Apr 25 2024 *)

{ for (n=0, 1000, write("b062988.txt", n, " ", binomial(n + 6, 5) - 1) ) } \\ Harry J. Smith, Aug 15 2009

[binomial(n+6,5) -1 for n in range(41)] # G. C. Greubel, Apr 25 2024

a(n) = A062985(n+2, 5).
a(n) = (n+1)*(n^4 + 19*n^3 + 136*n^2 + 444*n + 600)/5!.
G.f.: N(5;1, x)/(1-x)^6 with N(5;1, x)= 5 - 10*x + 10*x^2 - 5*x^3 + x^4 = (1-(1-x)^5)/x, polynomial of second row of A062986.
E.g.f.: (1/120)*(600 + 1800*x + 1200*x^2 + 300*x^3 + 30*x^4 + x^5)*exp(x). - G. C. Greubel, Apr 25 2024

A332662 Put-and-count: An enumeration of N X N where N = {0, 1, 2, ...}. The terms are interleaved x and y coordinates. Or: A row-wise storage scheme for sequences of regular triangles.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 1, 1, 2, 1, 3, 0, 4, 0, 5, 0, 0, 3, 1, 2, 2, 2, 3, 1, 4, 1, 5, 1, 6, 0, 7, 0, 8, 0, 9, 0, 0, 4, 1, 3, 2, 3, 3, 2, 4, 2, 5, 2, 6, 1, 7, 1, 8, 1, 9, 1, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 0, 5, 1, 4, 2, 4, 3, 3, 4, 3, 5, 3, 6, 2, 7, 2

Offset: 0

Peter Luschny, Feb 18 2020

nonn

Other enumerations of N X N designed with storage allocation for extensible arrays in mind include A319514 and A319571.

			Illustrating the linear storage layout of a sequence of regular triangles.
(A) [ 0], [ 2,  3], [ 7,  8,  9], [16, 17, 18, 19], [30, 31, 32, 33, 34], ...
(B) [ 1], [ 5,  6], [13, 14, 15], [26, 27, 28, 29], ...
(C) [ 4], [11, 12], [23, 24, 25], ...
(D) [10], [21, 22], ...
(E) [20], ...
...
The first column is A000292.
The start values of all partial rows (in ascending order) are 0 plus A014370.
The start values of the partial rows in the first row are A005581 (without first 0).
The start values of the partial rows on the main diagonal are A331987.
The end values of all partial rows (in ascending order) are A332023.
The end values of the partial rows in the first row are A062748.
The end values of the partial rows on the main diagonal are A332698.

Peter Luschny, Table of n, a(n) for n = 0..10000

A332663 (x-coordinates), A056559 (y-coordinates).
Cf. A000292, A014370, A002260, A005581, A319514, A319571, A331987.
Cf. A332023, A062748, A332698.

function a_list(N)
    a = Int[]
    for n in 1:N
        i = 0
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, i, t)
                i += 1
end end end; a end
a_list(5) |> println

count := (k, A) -> ListTools:-Occurrences(k, A): t := n -> n*(n+1)/2:
PutAndCount := proc(N) local L, n, v, c, seq; L := NULL; seq := NULL;
for n from 1 to N do
   for v from 0 to t(n)-1 do
     # How often did you see v in this sequence before?
     c := count(v, [seq]);
     L := L, v, c; seq := seq, v;
od od; L end:  PutAndCount(6);
# Returning 'seq' instead of 'L' gives the x-coordinates (A332663).

t[n_] := n*(n+1)/2;
PutAndCount[N_] := Module[{L, n, v, c, seq},
L = {}; seq = {};
For[n = 1, n <= N, n++,
   For[v = 0, v <= t[n]-1, v++,
      c = Count[seq, v];
      L = Join[L, {v, c}]; seq = Append[seq, v]
]]; L];
PutAndCount[6] (* Jean-François Alcover, Oct 13 2024, after Maple program *)

A118714 Determinant of n X n matrix whose diagonal contains the first n tetrahedral numbers and all other elements are 1's.

Original entry on oeis.org

1, 3, 27, 513, 17442, 959310, 79622730, 9475104870, 1553917198680, 340307866510920, 96987741955612200, 35206550329887228600, 15983773849768801784400, 8934929582020760197479600, 6066817186192096174088648400, 4944456006746558381882248446000

Offset: 1

Alexander Adamchuk, May 20 2006

nonn

a(n+2) / a(n+1) = A062748(n) = A062745(n+2, 3)= binomial(n+4, 3)-1 = (n+1)*(n^2+8*n+18)/3!.

			The matrix begins:
1 1 1 1 1 1 1 ...
1 4 1 1 1 1 1 ...
1 1 10 1 1 1 1 ...
1 1 1 20 1 1 1 ...
1 1 1 1 35 1 1 ...
1 1 1 1 1 56 1 ...

Colin Barker, Table of n, a(n) for n = 1..150

Cf. A000292, A067550, A062748, A062745.

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1) *(6+4*n+n^2)*(n-1)/6)
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 15 2015

Table[ Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ], {n, 1, 20} ]
Table[Product[(k-3)*(k^2+2)/3!,{k,4,n+2}],{n,1,20}]

a(n) = matdet(matrix(n, n, i, j, if(i==j, i*(i+1)*(i+2)/6, 1))) \\ Colin Barker, Nov 13 2015

a(n) = Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ].
a(n) = Product[(j-3)*(j^2+2)/3!,{j,4,n+2}].
a(n) = Product[(k+1)*(k^2+8*k+18)/3!,{k,0,n-2}] = Product[A062748(k),{k,0,n-2}].
a(n) ~ sqrt(Pi) * sinh(Pi*sqrt(2)) * n^(3*n + 9/2) / (11 * 2^(n-1) * 3^(n+1) * exp(3*n)). - Vaclav Kotesovec, Apr 17 2018

a(15) and a(16) from Colin Barker, Nov 13 2015

A124089 a(n) = binomial(n,6)-1.

Original entry on oeis.org

0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, 18563, 27131, 38759, 54263, 74612, 100946, 134595, 177099, 230229, 296009, 376739, 475019, 593774, 736280, 906191, 1107567, 1344903, 1623159, 1947791, 2324783, 2760680, 3262622

Offset: 6

Zerinvary Lajos, Nov 25 2006

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000096, A000579, A062748, A063258, A062988.

[Binomial(n,6)-1 : n in [6..40]]; // Wesley Ivan Hurt, Dec 27 2023

Maple
```
[seq(binomial(n,6)-1,n=6..42)];
```

Binomial[Range[6,40],6]-1 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,27,83,209,461,923},40] (* Harvey P. Dale, Dec 26 2015 *)

a(n) = A000579(n)-1.

a(0)=0, a(1)=6, a(2)=27, a(3)=83, a(4)=209, a(5)=461, a(6)=923, a(n)= 7*a(n-1)- 21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+ a(n-7). - Harvey P. Dale, Dec 26 2015

A332023 T(n, k) = binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 12, 15, 19, 20, 22, 25, 29, 34, 35, 37, 40, 44, 49, 55, 56, 58, 61, 65, 70, 76, 83, 84, 86, 89, 93, 98, 104, 111, 119, 120, 122, 125, 129, 134, 140, 147, 155, 164, 165, 167, 170, 174, 179, 185, 192, 200, 209, 219

Offset: 0

Peter Luschny, Feb 20 2020

The sequence increases monotonically.

			The triangle starts:
[0]   0;
[1]   1,   3;
[2]   4,   6,   9;
[3]  10,  12,  15,  19;
[4]  20,  22,  25,  29,  34;
[5]  35,  37,  40,  44,  49,  55;
[6]  56,  58,  61,  65,  70,  76,  83;
[7]  84,  86,  89,  93,  98, 104, 111, 119;
[8] 120, 122, 125, 129, 134, 140, 147, 155, 164;
[9] 165, 167, 170, 174, 179, 185, 192, 200, 209, 219;

Cf. A000292 (first column), A062748 (diagonal), A005286 (subdiagonal), A332697 (row sums).

Cf. A014370.

T := (n, k) -> binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1):
seq(seq(T(n, k), k=0..n), n=0..9);

T(n, k) = (1/6)*(3*k^2 + 9*k + n*(n + 1)*(n + 2)).

cp's OEIS Frontend

A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

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A063258 a(n) = binomial(n+5,4) - 1.

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A062745 Generalized Catalan array FS(3; n,r).

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A220032 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or diagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..1 nXk array.

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A014430 Subtract 1 from Pascal's triangle, read by rows.

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A062988 a(n) = binomial(n+6,5) - 1.

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