A062988 -id:A062988 - 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

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

A062989 a(n) = C(n+6, 6) - n - 1.

Original entry on oeis.org

0, 5, 25, 80, 205, 456, 917, 1708, 2994, 4995, 7997, 12364, 18551, 27118, 38745, 54248, 74596, 100929, 134577, 177080, 230209, 295988, 376717, 474996, 593750, 736255, 906165, 1107540, 1344875, 1623130, 1947761, 2324752, 2760648, 3262589, 3838345, 4496352

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{4}, n >= 0.

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

Seventh column (r=6) of FS(5) staircase array A062985.

Partial sums of A062988.

Table[Binomial[n+6,6]-n-1,{n,0,40}] (* OR *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,5,25,80,205,456,917},40] (* Harvey P. Dale, Aug 08 2013 *)

{ for (n=0, 1000, write("b062989.txt", n, " ", binomial(n + 6, 6) - n - 1) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062985(n+2, 6) = (n+1)*(n+2)*(n^4 + 24*n^3 + 221*n^2 + 954*n + 1800)/6!.
G.f.: N(5;1, x)/(1-x)^7 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.
a(0)=0, a(1)=5, a(2)=25, a(3)=80, a(4)=205, a(5)=456, a(6)=917, 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, Aug 08 2013
D-finite with recurrence -n*a(n) +(n+6)*a(n-1) +5*n=0. - R. J. Mathar, Nov 22 2024

Simpler definition from Zerinvary Lajos, May 08 2006

A062985 Generalized Catalan array FS(5; n,r).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 1, 3, 6, 10, 15, 20, 25, 30, 35, 35, 35, 35, 35, 1, 4, 10, 20, 35, 55, 80, 110, 145, 180, 215, 250, 285, 285, 285, 285, 285, 1, 5, 15, 35, 70, 125, 205, 315, 460, 640, 855, 1105

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this array is called {n over r}_{m-1}, with m=5.
The step width sequence of this staircase array is [1,4,4,4,....], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.
The columns r=0..7 (without leading zeros) give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A000332(4+n), A062988-A062990.

			{1}; {1,1,1,1,1}; {1,2,3,4,5,5,5,5,5}; ...; N(5; 1,x)=5-10*x+10*x^2-5*x^3+x^4.

D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.

a(0, 0)=1, a(n, -1)=0, n >= 1; a(n, r)=0 if r>4*n; a(n, r)=a(n, r-1)+a(n-1, r) else.

G.f. for column r=4*k+j, k >= 0, j=1, 2, 3, 4: (x^(k+1))*N(5; k, x)/(1-x)^(4*k+1+j), with row polynomials of array A062986.

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

A124090 C(n,7)-1.

Original entry on oeis.org

0, 7, 35, 119, 329, 791, 1715, 3431, 6434, 11439, 19447, 31823, 50387, 77519, 116279, 170543, 245156, 346103, 480699, 657799, 888029, 1184039, 1560779, 2035799, 2629574, 3365855, 4272047, 5379615, 6724519, 8347679, 10295471, 12620255

Offset: 7

Zerinvary Lajos, Nov 25 2006

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

Cf. A000096, A062748, A063258, A062988.

Maple
```
[seq(binomial(n,7)-1,n=7..47)];
```

Binomial[Range[7,50],7]-1 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,7,35,119,329,791,1715,3431},40] (* Harvey P. Dale, Aug 14 2014 *)

A165618 a(n) = binomial(n+8,8) - 1.

Original entry on oeis.org

0, 8, 44, 164, 494, 1286, 3002, 6434, 12869, 24309, 43757, 75581, 125969, 203489, 319769, 490313, 735470, 1081574, 1562274, 2220074, 3108104, 4292144, 5852924, 7888724, 10518299, 13884155, 18156203, 23535819, 30260339, 38608019, 48903491

Offset: 0

Enrique Pérez Herrero, Sep 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Pérez Herrero, Binomial Matrix (I) partitions, Psychedelic Geometry Blogspot, 09/22/09
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000096, A000581, A062748, A063258, A062988, A124089, A124090, A035927.

Table[ -1 + Binomial[n + 8, 8], {n, 0, 30}]
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,8,44,164,494,1286,3002,6434,12869},40] (* Harvey P. Dale, Nov 18 2013 *)

vector(100,n,binomial(n+7,8)-1) \\ Charles R Greathouse IV, May 27 2011

a(n) = binomial(n+8,8) - 1 = A000581(n+8) - 1.
a(n) = Sum_{r=1..n} binomial(8,r)*binomial(n,r).
a(n) = n(n+9)(n^6 + 27n^5 + 303n^4 + 1809n^3 + 6168n^2 + 11772n + 12176)/40320.

Edited by Charles R Greathouse IV, May 27 2011

cp's OEIS Frontend

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

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

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A062989 a(n) = C(n+6, 6) - n - 1.

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A062985 Generalized Catalan array FS(5; n,r).

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

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A124090 C(n,7)-1.

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A165618 a(n) = binomial(n+8,8) - 1.

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