A080956 -id:A080956 - cp's OEIS frontend

A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 15, 8, 1, 1, 1, 5, 34, 105, 16, 1, 1, 1, 6, 61, 496, 945, 32, 1, 1, 1, 7, 96, 1385, 11056, 10395, 64, 1, 1, 1, 8, 139, 2976, 50521, 349504, 135135, 128, 1, 1, 1, 9, 190, 5473, 151416, 2702765, 14873104, 2027025, 256, 1

Offset: 0

Peter Luschny, Sep 30 2023

Using polygonal numbers as weights, a recursion for triangles is defined, whose main diagonals represents a family of sequences, which include, among others, the powers of 2, the double factorial of odd numbers, the reduced tangent numbers, and the Euler numbers.
Apart from the edge cases k = 0 and k = n the recursion is T(n, k) = w(n, k) * T(n, k - 1) + T(n - 1, k). T(n, 0) = 1 and T(n, n) = T(n, n-1) if n > 0.
The weights w(n, k) identical to 1 yield the recursion of the Catalan triangle A009766 (with main diagonal the Catalan numbers). Here the polygonal numbers are used as weights in the form w(n, k) = p(s, n - k + 1), where the parameter s is the number of sides of the polygon and p(s, n) = ((s-2) * n^2 - (s-4) * n) / 2, see A317302.

			Array A(n, k) starts:                            (polygon|diagonal|triangle)
[0] 1, 1, 1,   1,     1,       1,         1, ...  A258837  A000012
[1] 1, 1, 2,   4,     8,      16,        32, ...  A080956  A011782
[2] 1, 1, 3,  15,   105,     945,     10395, ...  A001477  A001147  A001498
[3] 1, 1, 4,  34,   496,   11056,    349504, ...  A000217  A002105  A365674
[4] 1, 1, 5,  61,  1385,   50521,   2702765, ...  A000290  A000364  A060058
[5] 1, 1, 6,  96,  2976,  151416,  11449296, ...  A000326  A126151  A366138
[6] 1, 1, 7, 139,  5473,  357721,  34988647, ...  A000384  A126156  A365672
[7] 1, 1, 8, 190,  9080,  725320,  87067520, ...  A000566  A366150  A366149
[8] 1, 1, 9, 249, 14001, 1322001, 188106489, ...  A000567
           A054556                         A366137

Wikipedia, Polygonal number.

Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384.
Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150.
Triangles: A001498, A365674, A060058, A366138, A365672, A366149.
Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers).
Cf. A112934, A303943, A305532, A305533.

poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:
T := proc(s, n, k) option remember; if k = 0 then 1 else if k = n then T(s, n, k-1) else poly(s, n - k + 1) * T(s, n, k - 1) + T(s, n - 1, k) fi fi end:
for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;
# Alternative, using continued fractions:
A := proc(p, L) local CF, poly, k, m, P, ser;
   poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;
   CF := 1 + x;
   for k from 1 to L do
       m := L - k + 1;
       P := poly(p, m);
       CF := 1/(1 - P*x*CF)
   od;
   ser := series(CF, x, L);
   seq(coeff(ser, x, m), m = 0..L-1)
end:
for p from 0 to 8 do lprint(A(p, 8)) od;

poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;
T[s_, n_, k_] := T[s, n, k] = If[k == 0, 1, If[k == n, T[s, n, k - 1], poly[s, n - k + 1] * T[s, n, k - 1] + T[s, n - 1, k]]];
A[n_, k_] := T[n, k, k];
Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2023, from first Maple program *)

A(p, n) = {
       my(CF = 1 + x,
           poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,
           m, P
       );
       for(k = 1, n,
           m = n - k + 1;
           P = poly(p, m);
           CF = 1/(1 - P*x*CF)
        );
        Vec(CF + O(x^(n)))
}
for(p = 0, 8, print(A(p, 8)))
\\  Michel Marcus and Peter Luschny, Oct 02 2023

from functools import cache
@cache
def T(s, n, k):
    if k == 0: return 1
    if k == n: return T(s, n, k - 1)
    p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2
    return p * T(s, n, k - 1) + T(s, n - 1, k)
def A(n, k): return T(n, k, k)
for n in range(9): print([A(n, k) for k in range(9)])

A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

Original entry on oeis.org

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0

Offset: 0

Paul Curtz, Jun 24 2011

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1,  -1,  -1,   -1,
1,  0, -1, -2, -3,  -4,  -5,   -6,
1,  1,  0, -2, -5,  -9, -14,  -20,
1,  2,  2,  0, -5, -14, -28,  -48,
1,  3,  5,  5,  0, -14, -42,  -90,
1,  4,  9, 14, 14,   0, -42, -132,
1,  5, 14, 28, 42,  42,   0, -132,
1,  6, 20, 48, 90, 132, 132,    0.
The rows in this array are essentially the columns of T(n,k).

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66

Offset: 0

Omar E. Pol, Aug 09 2018

Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the nonnegative numbers.
For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------------
n\k  Numbers       Seq. No.   0   1   2   3   4    5    6    7    8
------------------------------------------------------------------------
0    ............ (A258837):  0,  1,  0, -3, -8, -15, -24, -35, -48, ...
1    ............ (A080956):  0,  1,  1,  0, -2,  -5,  -9, -14, -20, ...
2    Nonnegatives  A001477:   0,  1,  2,  3,  4,   5,   6,   7,   8, ...
3    Triangulars   A000217:   0,  1,  3,  6, 10,  15,  21,  28,  36, ...
4    Squares       A000290:   0,  1,  4,  9, 16,  25,  36,  49,  64, ...
5    Pentagonals   A000326:   0,  1,  5, 12, 22,  35,  51,  70,  92, ...
6    Hexagonals    A000384:   0,  1,  6, 15, 28,  45,  66,  91, 120, ...
7    Heptagonals   A000566:   0,  1,  7, 18, 34,  55,  81, 112, 148, ...
8    Octagonals    A000567:   0,  1,  8, 21, 40,  65,  96, 133, 176, ...
9    9-gonals      A001106:   0,  1,  9, 24, 46,  75, 111, 154, 204, ...
10   10-gonals     A001107:   0,  1, 10, 27, 52,  85, 126, 175, 232, ...
11   11-gonals     A051682:   0,  1, 11, 30, 58,  95, 141, 196, 260, ...
12   12-gonals     A051624:   0,  1, 12, 33, 64, 105, 156, 217, 288, ...
13   13-gonals     A051865:   0,  1, 13, 36, 70, 115, 171, 238, 316, ...
14   14-gonals     A051866:   0,  1, 14, 39, 76, 125, 186, 259, 344, ...
15   15-gonals     A051867:   0,  1, 15, 42, 82, 135, 201, 280, 372, ...
...

Omar E. Pol, Polygonal numbers.
The OEIS, Polygonal numbers.
University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.
Eric Weisstein's World of Mathematics, Figurate Number.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Column 0 gives A000004.
Column 1 gives A000012.
Column 2 gives A001477, which coincides with the row numbers.
Main diagonal gives A060354.
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the same as column 2.
For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).
Cf. A139600, A139601.
Cf. A303301 (similar table but with generalized polygonal numbers).

T(n,k) = A139600(n-2,k) if n >= 2.

T(n,k) = A139601(n-3,k) if n >= 3.

A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 2, 0, -2, 1, 1, -3, 2, 2, -3, 1, -1, 4, -5, 0, 5, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, -1, 6, -14, 14, 0, -14, 14, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, -1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, -1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Inverse is A112465.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 07 2006; corrected by Philippe Deléham, Dec 11 2008
Equals A097808 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle starts
   1;
   1,  1;
  -1,  0,  1;
   1, -1, -1,  1;
  -1,  2,  0, -2,  1;
   1, -3,  2,  2, -3,  1;
  -1,  4, -5,  0,  5, -4,  1;
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1,  1;
  -2, -1,  1;
   2,  0, -1,  1;
  -2,  0,  0, -1,  1;
   2,  0,  0,  0, -1,  1;
  -2,  0,  0,  0,  0, -1,  1;
   2,  0,  0,  0,  0,  0, -1,  1; (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

Cf. A000007, A008482, A037012, A057079, A097808, A112467.

Columns: A248157(n+2) (k=1), (-1)^n*A080956(n-2) (k=2), (-1)^(n-1)*A254749(n-2) (k=3).

A112466:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(n,k) - 2*Binomial(n-1,k)) >;
[A112466(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2025

seq(seq( (-1)^(n-k)*(2*binomial(n-1, k-1)-binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Feb 19 2020

{1}~Join~Table[(Binomial[n, n - k] - 2 Binomial[n - 1, n - k - 1])*(-1)^(n - k), {n, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 18 2020 *)

T(n,k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ Michel Marcus, Feb 19 2020

def A112466(n,k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n,k) - 2*binomial(n-1,k))
print(flatten([[A112466(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 30 2025

Number triangle: T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)), with T(0,0) = 1.
T(2*n, n) = 0 (main diagonal).
Sum_{k=0..n} T(n, k) = 0 + [n=0] + 2*[n=1] (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*Fibonacci(n-2) (diagonal sums).
Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - Philippe Deléham, Oct 03 2005
T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if n < 0 or if n < k, T(n,k) = T(n-1,k-1) - T(n-1,k) for n > 1. - Philippe Deléham, Nov 26 2006
G.f.: (1+2*x)/(1+x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Apr 30 2025: (Start)
T(2*n+1, 2*n+1-k) = T(2*n+1, k) (symmetric odd n rows).
T(2*n, 2*n-k) = (-1)*T(2*n, k) (antisymmetric even n rows).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) (signed row sums).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A057079(n+2) (signed diagonal sums). (End)

A176239 Shifted signed Catalan triangle T(n,k) = (-1)^(n+k+1)A009766(n,k-n+1) read by rows.

Original entry on oeis.org

0, -1, 1, -1, 0, 2, 0, 1, -2, 2, 0, -5, 0, 0, 1, -3, 5, -5, 0, 14, 0, 0, 0, 1, -4, 9, -14, 14, 0, -42, 0, 0, 0, 0, 1, -5, 14, -28, 42, -42, 0, 132, 0, 0, 0, 0, 0, 1, -6, 20, -48, 90, -132, 132, 0, -429, 0, 0, 0, 0, 0, 0, 1, -7, 27, -75, 165, -297, 429, -429, 0, 1430

Offset: 0

Paul Curtz, Apr 12 2010

			The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
0,-1;                          (-1)^k*k  A001477
1,-1,.0,.2;                      (-1)^(k+1)*(k+1)*(k-2)/2  A080956, A000096
0,.1,-2,.2,.0,-5;                 (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
0,.0,.1,-3,.5,-5,..0,.14;             (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
0,.0,.0,.1,-4,.9,-14,.14,.0,-42;           A005557, A034807
0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;

Cf A140344, A033184.

A009766 := proc(n,k) if k<0 or k >n then 0; else binomial(n+k,n)*(n-k+1)/(n+1) ; end if; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A176239 := proc(n,k) if k <= 2*n-1 then (-1)^(n+k+1)*A009766(n,k-n+1) elif k = 2*n then 0; elif k < 2*(n+1) then (-1)^(n+1)*A000108(n+1); else 0; end if; end proc: # R. J. Mathar, Dec 03 2010

T(n,k) = T(n+1,k)+T(n+1,k+1), k <= 2n+1.
T(n,2n) = 0.
T(n,2n+1) = (-1)^(n+1)*A000108(n+1).
T(n,k) =  (-1)^(n+k+1)*A009766(n,k-n+1), k < 2n.

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A080959 Square array of coefficients of binomial polynomials, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 4, 0, 11, 14, 5, -2, 20, 14, 94, 6, -5, 34, -10, 214, 444, 7, -9, 55, -74, 454, 444, 3828, 8, -14, 85, -200, 974, -636, 8868, 25584, 9, -20, 126, -416, 2024, -4236, 21468, 25584, 270576, 10, -27, 180, -756, 3968, -13056, 56748, -55056, 633456, 2342880, 11, -35, 249, -1260, 7308, -31632, 146208, -377616, 1722096, 2342880, 29400480

Offset: 1

Paul Barry, Mar 01 2003

			Array, A(n, k), begin as:
   1,   1,   5,    14,    94,     444,    3828,      25584,     270576, ... A024167;
   2,   1,  11,    14,   214,     444,    8868,      25584,     633456, ... A080958;
   3,   0,  20,   -10,   454,    -636,   21468,     -55056,    1722096, ... ;
   4,  -2,  34,   -74,   974,   -4236,   56748,    -377616,    5471856, ... ;
   5,  -5,  55,  -200,  2024,  -13056,  146208,   -1325136,   16902576, ... ;
   6,  -9,  85,  -416,  3968,  -31632,  348816,   -3695952,   47457072, ... ;
   7, -14, 126,  -756,  7308,  -67032,  766296,   -9004752,  120758832, ... ;
   8, -20, 180, -1260, 12708, -129672, 1563336,  -19925712,  281929392, ... ;
   9, -27, 249, -1974, 21018, -234252, 2993436,  -40917312,  611923392, ... ;
  10, -35, 335, -2950, 33298, -400812, 5431116,  -79073472, 1248697152, ... ;
  11, -44, 440, -4246, 50842, -655908, 9411204, -145250688, 2417424768, ... ;
Antidiagonals, T(n, k), begin as:
   1;
   2,   1;
   3,   1,   5;
   4,   0,  11,    14;
   5,  -2,  20,    14,   94;
   6,  -5,  34,   -10,  214,    444;
   7,  -9,  55,   -74,  454,    444,   3828;
   8, -14,  85,  -200,  974,   -636,   8868,   25584;
   9, -20, 126,  -416, 2024,  -4236,  21468,   25584,  270576;
  10, -27, 180,  -756, 3968, -13056,  56748,  -55056,  633456, 2342880;
  11, -35, 249, -1260, 7308, -31632, 146208, -377616, 1722096, 2342880, 29400480;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Cf. A024167, A080958.

Columns: A000027 (k=1), A080956 (k=2), A080957 (k=3).

A:= func< n,k | Factorial(k)*(&+[(-1)^(j+1)*Binomial(n+j,j)/j: j in [1..k]]) >;
A080959:= func< n,k | A(n-k,k) >;
[A080959(n,k): k in [1..n], n in [0..12]]; // G. C. Greubel, May 11 2025

A[n_, k_]:= k!*Sum[(-1)^(j+1)*Binomial[n+j,j]/j, {j,k}];
A080959[n_, k_]:= A[n-k, k];
Table[A080959[n,k], {n,0,12}, {k,n}]//Flatten (* G. C. Greubel, May 11 2025 *)

def A(n,k): return factorial(k)*sum((-1)^(j+1)*binomial(n+j,j)/j for j in range(1,k+1))
def A080959(n,k): return A(n-k,k)
print(flatten([[A080959(n,k) for k in range(1,n+1)] for n in range(13)])) # G. C. Greubel, May 11 2025

A(n, k) = k!*Sum_{j=1..k} (-1)^(j+1)*binomial(n+j, j)/j (array).

T(n, k) = A(n-k, k) (antidiagonals).

A319930 a(n) = (1/24)n(n - 1)(n - 3)(n - 14).

Original entry on oeis.org

0, 0, 1, 0, -5, -15, -30, -49, -70, -90, -105, -110, -99, -65, 0, 105, 260, 476, 765, 1140, 1615, 2205, 2926, 3795, 4830, 6050, 7475, 9126, 11025, 13195, 15660, 18445, 21576, 25080, 28985, 33320, 38115, 43401, 49210, 55575, 62530, 70110, 78351, 87290, 96965

Offset: 0

Peter Luschny, Oct 02 2018

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012 (m=0), A001489 (m=1), A080956 (m=2), A167541 (m=3), this sequence (m=4), A319931 (m=5), A319932 (m=6).

Cf. A319933.

a := n -> (1/24)*n*(n-1)*(n-3)*(n-14):
seq(a(n), n=0..44);

Table[(n(n-1)(n-3)(n-14))/24,{n,0,70}] (* Harvey P. Dale, Apr 29 2022 *)

a(n) = [x^4] DedekindEta(x)^n.

a(n) = A319933(n, 4).

A319931 a(n) = -(1/120)n(n - 3)(n - 6)(n^2 - 21*n + 8).

Original entry on oeis.org

0, 1, 2, 0, -4, -6, 0, 21, 64, 135, 238, 374, 540, 728, 924, 1107, 1248, 1309, 1242, 988, 476, -378, -1672, -3519, -6048, -9405, -13754, -19278, -26180, -34684, -45036, -57505, -72384, -89991, -110670, -134792, -162756, -194990, -231952, -274131, -322048, -376257

Offset: 0

Peter Luschny, Oct 02 2018

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000012 (m=0), A001489 (m=1), A080956 (m=2), A167541 (m=3), A319930 (m=4), this sequence (m=5), A319932 (m=6).

Cf. A319933.

a := n -> -(1/120)*n*(n-3)*(n-6)*(n^2-21*n+8):
seq(a(n), n=0..41);

a(n)=-n*(n-3)*(n-6)*(n^2-21*n+8)/120 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = [x^5] DedekindEta(x)^n.
a(n) = A319933(n, 5).
From Chai Wah Wu, Jul 27 2022: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 5.
G.f.: x*(-7*x^4 + 6*x^3 + 3*x^2 - 4*x + 1)/(x - 1)^6. (End)

A319932 a(n) = (1/720)n(n - 10)(n - 1)(n^3 - 34n^2 + 181n - 144).

Original entry on oeis.org

0, 0, -2, -7, -10, -5, 11, 35, 56, 54, 0, -143, -418, -871, -1547, -2485, -3712, -5236, -7038, -9063, -11210, -13321, -15169, -16445, -16744, -15550, -12220, -5967, 4158, 19285, 40745, 70091, 109120, 159896, 224774, 306425, 407862, 532467, 684019, 866723

Offset: 0

Peter Luschny, Oct 02 2018

sign

Cf. A000012 (m=0), A001489 (m=1), A080956 (m=2), A167541 (m=3), A319930 (m=4), A319931 (m=5), this sequence (m=6).

Cf. A319933.

a := n -> (1/720)*n*(n-10)*(n-1)*(n^3-34*n^2+181*n-144);
seq(a(n), n=0..39);

a(n) = [x^5] DedekindEta(x)^n.

a(n) = A319933(n, 5).

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A080959 Square array of coefficients of binomial polynomials, read by antidiagonals.

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A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

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