A002414 -id:A002414 - cp's OEIS frontend

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

1, 5, 4, 12, 11, 7, 22, 21, 17, 10, 35, 34, 30, 23, 13, 51, 50, 46, 39, 29, 16, 70, 69, 65, 58, 48, 35, 19, 92, 91, 87, 80, 70, 57, 41, 22, 117, 116, 112, 105, 95, 82, 66, 47, 25, 145, 144, 140, 133, 123, 110, 94, 75, 53, 28, 176, 175, 171, 164, 154, 141, 125, 106, 84, 59

Offset: 1

Gary W. Adamson, Jun 10 2004, Jul 28 2008

Octagonal pyramidal number triangle, read by rows.
The triangle is generated from the product B*A of the infinite lower triangular matrices A =
  1 0 0 0...
  1 1 0 0...
  1 1 1 0...
  1 1 1 1...
and B =
  1 0 0 0...
  1 4 0 0...
  1 4 7 0...
  1 4 7 10...
T(n,0)=A000326(n+1)
T(n,2)=A059845(n+2)
T(n,n)=3*n+1

			Column 3 = A059845: 7, 17, 30, 46, 65...; while rightmost terms of rows are 1, 4, 7, 10...
First few rows of the triangle =
  1;
  5, 4;
  12, 11, 7;
  22, 21, 17, 10;
  35, 34, 30, 23, 13;
  51, 50, 46, 39, 29, 16;
  70, 69, 65, 58, 48, 35, 19;
  ...

Cf. A095872, A000326, A059845, A002414 (row sums)

T(n, k) = local(i); if(k>n,0,(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2)
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print()) \\ Lambert Klasen

Triangle read by rows, T(n,k) = sum {j=k..n} 3*j - 2 = A000012 * ((3*j - 2) * 0^(n-k)) * A000012; 1<=k<=n. E.g. T(5,3) = 30 = (7 + 10 + 13).

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 21 2005

A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.

Original entry on oeis.org

1, 5, 16, 12, 44, 49, 22, 84, 119, 100, 35, 136, 210, 230, 169, 51, 200, 322, 390, 377, 256, 70, 176, 455, 580, 624, 560, 361, 70, 276, 455, 580, 624, 560, 361, 92, 364, 609, 800, 910, 912, 779, 484, 117, 464, 784, 1050, 1235, 1312, 1254, 1034, 625, 145, 576, 980, 1330, 1599

Offset: 1

Gary W. Adamson, Jun 10 2004

Arranged by flush left columns (k=1,2,3...), (k=1) column = A000326, the pentagonal numbers (1, 5, 12, 22, 35...). The Octagonal pyramidal number triangle of A095871 is generated from A095872 by dividing the k-th row by the n-th term in the series 1, 4, 7, 10...(k starting with 1). Dividing the 3rd column of A095872 (49, 119, 210, 322, 455...) by 7 generates A059845: 7, 17, 30, 46, 65... Rightmost terms of each row of A095872 are A016778 (1, 16, 49, 100, 169...); i.e. squares of 1, 4, 7, 10... Row sums of A095872 are 1, 21, 105, 325, 780, 1596, 2926... Row sums of A095871 are the octagonal pyramidal numbers, A002414: 1, 9, 30, 70, 135, 231, 364...

[Editor's note: OEIS' "TABL" format (fmt=2) rather displays the transposed matrix as upper triangular matrix.]

			Let M = the infinite lower triangular matrix in the format exemplified by a 3rd order matrix: [1 0 0 / 1 4 0 / 1 4 7]: i.e. for the n-th order matrix, each row has n terms in the series 1, 4, 7, 10... with the rest of the spaces filled in with zeros. Square the matrix and delete the zeros; then read by rows.
[1 0 0 / 1 4 0 / 1 4 7]^2 = [1 0 0 / 5 16 0 / 12 44 49]; then delete the zeros and read by rows: 1, 5, 16, 12, 44, 49...

Cf. A095871, A000326, A059845, A002414, A016778.

A095802(n)={ my( r=sqrtint(2*n)+1, T=matrix(r,r,i,j,if(j>=i,3*j-2))^2); concat(vector(#T,i,vecextract(T[,i],2^i-1)))[n] } \\ M. F. Hasler, Apr 18 2009

a(k(k+1)/2) = (3k-2)^2 (diagonal elements: squares of the initial series), a(k(k-1)/2+1) = A000326(k) (1st column: pentagonal numbers). - M. F. Hasler, Apr 18 2009

Edited and extended by M. F. Hasler, Apr 18 2009

A104728 Triangle T(n,k) = (k-1-n)(k-2-n)(k-2+2*n)/2 read by rows, 1<=k<=n.

Original entry on oeis.org

1, 9, 4, 30, 18, 7, 70, 48, 27, 10, 135, 100, 66, 36, 13, 231, 180, 130, 84, 45, 16, 364, 294, 225, 160, 102, 54, 19, 540, 448, 357, 270, 190, 120, 63, 22, 765, 648, 532, 420, 315, 220, 138, 72, 25, 1045, 900, 756, 616, 483, 360, 250, 156, 81, 28, 1386, 1210, 1035, 864, 700, 546, 405, 280, 174, 90, 31

Offset: 1

Gary W. Adamson, Mar 20 2005

The triangle is defined as the matrix product A * B, A = [1; 1, 4; 1, 4, 7;...]; B = [1; 2, 1; 3, 2, 1;...]; both infinite lower triangular matrices with the rest of the terms zeros.

			The first few rows of the triangle are:
1;
9,    4;
30,   18,   7;
70,   48,   27,   10;
135,  100,  66,   36,   13;
231,  180,  130,  84,   45,  16;
364,  294,  225,  160,  102, 54,  19;
540,  448,  357,  270,  190, 120, 63,  22;
765,  648,  532,  420,  315, 220, 138, 72,  25;
1045, 900,  756,  616,  483, 360, 250, 156, 81,  28;
1386, 1210, 1035, 864,  700, 546, 405, 280, 174, 90,  31;
1794, 1584, 1375, 1170, 972, 784, 609, 450, 310, 192, 99, 34, etc.

Cf. A051798 (row sums), A007586, A002414 (column 1).

A104728 := proc(n)
        (k-1-n)*(k-2-n)*(k-2+2*n)/2 ;
end proc:
seq(seq(A104728(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 07 2011

Table[(k-1-n)(k-2-n)(k-2+2n)/2,{n,20},{k,n}]//Flatten (* Harvey P. Dale, Dec 25 2018 *)

Name contributed by R. J. Mathar, Nov 07 2011

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

Original entry on oeis.org

1, 0, 1, -6, 0, 6, 0, -42, 0, 36, 36, 0, -288, 0, 216, 0, 468, 0, -1944, 0, 1296, -216, 0, 4536, 0, -12960, 0, 7776, 0, -4104, 0, 38880, 0, -85536, 0, 46656, 1296, 0, -51840, 0, 311040, 0, -559872, 0, 279936, 0, 32400, 0, -544320, 0, 2379456, 0, -3639168, 0, 1679616

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
     1;
     0,     1;
    -6,     0,      6;
     0,   -42,      0,    36;
    36,     0,   -288,     0,    216;
     0,   468,      0, -1944,      0,   1296;
  -216,     0,   4536,     0, -12960,      0,    7776;
     0, -4104,      0, 38880,      0, -85536,       0, 46656;
  1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

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

Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;
A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;
[A136526(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2022

(* First program *)
a= (b+1)/(b-1); c=0; b=2;
B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
(* Second program *)
B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);
Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)

def f(n,k):
    if (k==0): return (-1)^(n//2)
    else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)
def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)
flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 22 2022

T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.
From G. C. Greubel, Sep 22 2022: (Start)
T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).
T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.
T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).
T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.
T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.
T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.
T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.
T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.
T(n, n) = 6^(n-1) + (5/6)*[n=0].
T(n, n-2) = -6*A081106(n-2), n >= 2.
Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].
G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

Edited by G. C. Greubel, Sep 22 2022

A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.

Original entry on oeis.org

1, 2, 36, 300, 1960, 11340, 60984, 312312, 1544400, 7438860, 35103640, 162954792, 746347056, 3380195000, 15164074800, 67476121200, 298135873440, 1309153089420, 5717335239000, 24847720451400, 107520292479600, 463440029892840, 1990477619679120, 8521600803066000

Offset: 0

Alois P. Heinz, May 20 2016

a(0) = 1 by convention.

			a(1) = 2: 12, 21.
a(2) = 36: 1122, 1133, 1144, 1212, 1221, 1313, 1331, 1414, 1441, 2112, 2121, 2211, 2233, 2244, 2323, 2332, 2424, 2442, 3113, 3131, 3223, 3232, 3311, 3322, 3344, 3434, 3443, 4114, 4141, 4224, 4242, 4334, 4343, 4411, 4422, 4433.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A002414, A006752, A213820, A245687.

a:= proc(n) option remember; `if`(n<2, 2^n,
       2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)))
    end:
seq(a(n), n=0..30);

a[n_] := (2*n^3 + n^2 - n) * CatalanNumber[n]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Mar 12 2023 *)

G.f.: 1+(8*x+1)*2*x/(1-4*x)^(5/2).
a(n) = C(2*n,n)*C(2*n,2) for n>0, a(0)=1.
a(n) = 2*C(2*(n-1),n-1)*(2*n-1)^2, a(0)=1.
a(n) = 2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)) for n>1, a(n) = 2^n for n=0..1.
a(n) = A245687(2n,n).
a(n) = A000108(n)*A213820(n) = 2*A000108(n)*A002414(n) for n>0, a(0)=1.
Sum_{n>=0} 1/a(n) = 1 - log(sqrt(3)+2)*Pi/6 + 4*G/3, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 12 2023

A378361 Octagonal indices of numbers that are both octagonal and octagonal pyramidal.

Original entry on oeis.org

0, 1, 19, 45, 6413415, 16068720

Offset: 1

Kelvin Voskuijl, Nov 23 2024

			19 is a term because the 19th octagonal number (1045) is also the 10th octagonal pyramidal number.

Cf. A000567 (octagonal numbers), A002414 (octagonal pyramidal numbers).

Cf. A344376.

A125235 Triangle with the partial column sums of the octagonal numbers.

Original entry on oeis.org

1, 8, 1, 21, 9, 1, 40, 30, 10, 1, 65, 70, 40, 11, 1, 96, 135, 110, 51, 12, 1, 133, 231, 245, 161, 63, 13, 1, 176, 364, 476, 406, 224, 76, 14, 1, 225, 540, 840, 882, 630, 300, 90, 15, 1, 280, 765, 1380, 1722, 1512, 930, 390, 105, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

"Partial column sums" means the octagonal numbers are the 1st column, the 2nd column are the partial sums of the 1st column, the 3rd column are the partial sums of the 2nd, etc.

Row sums are 1, 9, 31, 81, 187, 405, 847 = 7*(2^n-1) - 6*n. - R. J. Mathar, Sep 06 2011

			First few rows of the triangle:
   1;
   8,   1;
  21,   9,   1;
  40,  30,  10,   1;
  65,  70,  40,  11,   1;
  96, 135, 110,  51,  12,   1;
  ...

Albert H. Beiler, Recreations in the Theory of Numbers, Dover (1966), p. 189.

Cf. A000567, A002414, A002419, A051843, A027810, A125232 - A125234.

t(n, k) = if (n <0, 0, if (k==1, n*(3*n-2), if (k > 1, t(n-1,k-1) + t(n-1,k))));
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Mar 04 2014

T(n,1) = A000567(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>1.
T(n,2) = A002414(n-1).
T(n,3) = A002419(n-2).
T(n,4) = A051843(n-4).
T(n,5) = A027810(n-6).

More terms from Michel Marcus, Mar 04 2014

A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

Original entry on oeis.org

40, 2100, 324000, 117771500, 86640153600, 115851776040000, 260111401804800000, 922852527136155000000, 4931966428685936640000000, 38193820496218904209973280000, 415101787718859995456102400000000

Offset: 3

Jonathan Vos Post, May 25 2008

The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:
===================================================================
..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|
k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047
k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|
k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|
k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|
k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|
k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|
===================================================================

			a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.

Eric W. Weisstein, Pyramidal Number

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A140729.

A130729 := proc(n) n!*(n+1)!*(n-2)^n*pochhammer(1+(5-n)/(n-2),n)/6^n ; end: seq(A130729(n),n=3..30) ; # R. J. Mathar, May 31 2008

A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].

a(n) ~ Pi^(3/2) * n^(4*n + 1/2) / (2^(n - 3/2) * 3^(n-1) * exp(3*n+2)) * (1 + (3*log(n) + 3*gamma + 5/4)/n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 29 2023

More terms from R. J. Mathar, May 31 2008

A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

Original entry on oeis.org

1, 2, 6, 3, 9, 18, 4, 12, 24, 40, 5, 15, 30, 50, 75, 6, 18, 36, 60, 90, 126, 7, 21, 42, 70, 105, 147, 196, 8, 24, 48, 80, 120, 168, 224, 288, 9, 27, 54, 90, 135, 189, 252, 324, 405, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
  1;
  2,  6;
  3,  9, 18;
  4, 12, 24, 40;
  5, 15, 30, 50,  75;
  6, 18, 36, 60,  90, 126;
  7, 21, 42, 70, 105, 147, 196;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000012, A127648, A127773.

Cf. A002024, A002411 (right border), A002414, A002417 (row sums), A011379.

[n*Binomial(k+1, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022

Table[n*Binomial[k+1, 2], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

flatten([[n*binomial(k+1, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.
Sum_{k=1..n} T(n, k) = A002417(n).
T(n, n) = A002411(n).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A002024(n,k) * A127773(n,k).
T(n, k) = n * binomial(k+1, 2).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/4)*(4*n - 3*floor((n+1)/2) + 3)*binomial(2 + floor((n+1)/2), 3).
T(2*n-1, n) = A002414(n), n >= 1.
T(2*n-2, n-1) = A011379(n-1), n >= 2. (End)

A172045 a(n) = (9n^4+10n^3-3n^2-4n)/12.

Original entry on oeis.org

0, 1, 17, 80, 240, 565, 1141, 2072, 3480, 5505, 8305, 12056, 16952, 23205, 31045, 40720, 52496, 66657, 83505, 103360, 126560, 153461, 184437, 219880, 260200, 305825, 357201, 414792, 479080, 550565, 629765, 717216, 813472, 919105, 1034705

Offset: 0

Vincenzo Librandi, Jan 24 2010

The sequence is related to A002414 (octagonal pyramidal numbers) by a(n) = n*A002414(n)-sum(A002414(i), i=1..n-1) for n>0.
This is the case d=3 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6)-sum(k*(k+1)*(2*d*k-2*d+3)/6, k=0..n-1) = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - Bruno Berselli, Nov 03 2010
Also, the sequence is related to A000567 by a(n) = sum( i*A000567(i), i=0..n ). [Bruno Berselli, Dec 19 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000567, A002414.

[(9*n^4+10*n^3-3*n^2-4*n)/12: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

CoefficientList[Series[x (1 + 12 x + 5 x^2)/(1 - x)^5,{x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,17,80,240},40] (* Harvey P. Dale, Aug 25 2019 *)

a(n) = n*(n+1)*(9*n^2+n-4)/12. - Bruno Berselli, Apr 21 2010

G.f. -x*(1 +12*x +5*x^2) / (x - 1)^5 . - R. J. Mathar, Nov 17 2011

Edited by Bruno Berselli, Oct 06 - 12 2010

cp's OEIS Frontend

A095871 Triangle read by rows: T(n,k)=(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2.

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A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.

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A104728 Triangle T(n,k) = (k-1-n)*(k-2-n)*(k-2+2*n)/2 read by rows, 1<=k<=n.

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A136526 Coefficients polynomials B(x, n) = ((1 + a + b)*x - c)*B(x, n-1) - a*b*B(x, n-2) with a = 3, b = 2, and c = 0.

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Magma

Mathematica

SageMath

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Extensions

A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.

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Maple

Mathematica

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A378361 Octagonal indices of numbers that are both octagonal and octagonal pyramidal.

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A125235 Triangle with the partial column sums of the octagonal numbers.

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PARI

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Extensions

A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

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A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

A104728 Triangle T(n,k) = (k-1-n)(k-2-n)(k-2+2*n)/2 read by rows, 1<=k<=n.

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

A172045 a(n) = (9n^4+10n^3-3n^2-4n)/12.