A034856 -id:A034856 - cp's OEIS frontend

A130297 A130296^2.

1, 4, 1, 8, 2, 1, 13, 3, 2, 1, 19, 4, 3, 2, 1, 26, 5, 4, 3, 2, 1, 34, 6, 5, 4, 3, 2, 1, 43, 7, 6, 5, 4, 3, 2, 1, 53, 8, 7, 6, 5, 4, 3, 2, 1, 64, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

Left border = A034856: (1, 4, 8, 13, 19, 26, 34, ...).

Row sums = A028387: (1, 5, 11, 19, 29, 41, 55, ...).

			First few rows of the triangle:
   1;
   4, 1;
   8, 2, 1;
  13, 3, 2, 1;
  19, 4, 3, 2, 1;
  26, 5, 4, 3, 2, 1;
  ...

Cf. A130296, A034856, A028387.

from math import comb, isqrt
def A130297(n): return (a:=comb(r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1))+1,2))+1-n+(a-1 if ((k2:=n-1<<1)==(m2:=isqrt(k2))*(m2+1)) else 0) # Chai Wah Wu, Nov 09 2024

Square of A130296 as an infinite lower triangular matrix.

A131416 (A000012 * A002260) + (A002260 * A000012) - A000012.

Original entry on oeis.org

1, 4, 3, 8, 8, 5, 13, 14, 12, 7, 19, 21, 20, 16, 9, 26, 29, 29, 26, 20, 11, 34, 38, 39, 37, 32, 24, 13, 43, 48, 50, 49, 45, 38, 28, 15, 53, 59, 62, 62, 59, 53, 44, 32, 17, 64, 71, 75, 76, 74, 69, 61, 50, 36, 19

Offset: 1

Gary W. Adamson, Jul 08 2007

Left column = A034856: (1, 4, 8, 13, 19, 26, 34,...). Row sums = A127735: (1, 7, 21, 46, 85...).

			First few rows of the triangle are:
1;
4, 3;
8, 8, 5;
13, 14, 12, 7;
19, 21, 20, 16, 9;
26, 29, 29, 26, 20, 11;
...

Cf. A002260, A034856, A127735, A000012.

(A000012 * A002260) + (A002260 * A000012) - A000012; as infinite lower triangular matrices.

A132118 Triangle read by rows: T(n,k) = n(n-1)/2 + 2k - 1.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 7, 9, 11, 13, 11, 13, 15, 17, 19, 16, 18, 20, 22, 24, 26, 22, 24, 26, 28, 30, 32, 34, 29, 31, 33, 35, 37, 39, 41, 43, 37, 39, 41, 43, 45, 47, 49, 51, 53, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76

Offset: 1

Gary W. Adamson, Aug 10 2007

			First few rows of the triangle are:
   1;
   2,  4;
   4,  6,  8;
   7,  9, 11, 13;
  11, 13, 15, 17, 19;
  16, 18, 20, 22, 24, 26;
  22, 24, 26, 28, 30, 32, 34;
  29, 31, 33, 35, 37, 39, 41, 43;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Column 1 is A000124(n-1).
Main diagonal is A034856.
Row sums are A002411.

Table[(n(n-1))/2+2k-1,{n,20},{k,n}]//Flatten (* Harvey P. Dale, Mar 26 2022 *)

a(16), a(17) corrected by Georg Fischer, Jul 01 2020

Name changed and terms a(56) and beyond from Andrew Howroyd, Apr 17 2021

A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 25, 13, 1, 1, 19, 59, 59, 19, 1, 1, 26, 119, 194, 119, 26, 1, 1, 34, 216, 524, 524, 216, 34, 1, 1, 43, 363, 1231, 1833, 1231, 363, 43, 1, 1, 53, 575, 2603, 5417, 5417, 2603, 575, 53, 1, 1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869

Offset: 1

Gary W. Adamson, Aug 30 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  4,   1;
  1,  8,   8,    1;
  1, 13,  25,   13,     1;
  1, 19,  59,   59,    19,     1;
  1, 26, 119,  194,   119,    26,     1;
  1, 34, 216,  524,   524,   216,    34,    1;
  1, 43, 363, 1231,  1833,  1231,   363,   43,   1;
  1, 53, 575, 2603,  5417,  5417,  2603,  575,  53,  1;
  1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869, 64, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Column k=2 is A034856.
Row sums are A132790.
Cf. A007318, A001263.

<< DiscreteMath`Combinatorica`
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
t[n_, m_, 2] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

T(n,k)={if(k<=n, binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1, 0)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 08 2018

Equals A007318 + A001263 - A000012 as infinite lower triangular matrices.
A symmetrical triangle recursion: let q=4; t(n,m,0)=Binomial[n,m]; t(n,m,1)=Narayana(n,m); t(n,m,2)=Eulerian(n+1,m); t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).
T(n,k) = binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1. - Andrew Howroyd, Sep 08 2018

More terms, Mma program and additional comments from Roger L. Bagula, Apr 20 2010
Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar
Name clarified by Andrew Howroyd, Sep 08 2018

A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 4, 8, 5, 1, 0, 4, 12, 13, 6, 1, 0, 4, 16, 25, 19, 7, 1, 0, 4, 20, 41, 44, 26, 8, 1, 0, 4, 24, 61, 85, 70, 34, 9, 1, 0, 4, 28, 85, 146, 155, 104, 43, 10, 1, 0, 4, 32, 113, 231, 301, 259, 147, 53, 11, 1, 0, 4, 36, 145, 344, 532, 560, 406, 200, 64, 12, 1

Offset: 0

Paul Curtz, Dec 16 2011

The array F(n,m), beginning with row n=0, is:
  1, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  1, 4,  8, 13,  19,  26,   34,
  0, 4, 12, 25,  44,  70,  104,
  0, 4, 16, 41,  85, 155,  259,
  0, 4, 20, 61, 146, 301,  560,
  0, 4, 24, 85, 231, 532, 1092.
Columns after A130713, A113311, A008574 have signatures (3,-3,1), (4,-6,4,-1), (5,-10,10,-5,1), (6,-15,20,-15,6,-1) (from A135278(n+3)).
Inserting columns of zeros and pushing the columns down, plus alternating sign switches defines the following triangle T(n,2m) = (-1)^(m/2)*F(n-2m,m):
  1,
  2 0,
  1 0 -1,
  0 0 -3 0,
  0 0 -4 0 1,
  0 0 -4 0 4 0,
  0 0 -4 0 8 0 -1
The row sums in the triangle are (-1)^n*A099838(n).
The companion to A201863 is
  1
  1 0
  1 0   0
  1 0  -2 0
  1 0  -4 0  1
  1 0  -6 0  5 0
  1 0  -8 0 13 0   -2
  1 0 -10 0 25 0  -12 0
  1 0 -12 0 41 0  -38 0   4
  1 0 -14 0 61 0  -88 0  28 0
  1 0 -16 0 85 0 -170 0 104 0 -8
5th column: A001844; 7th column: -A035597=-2*A005900(n+1); 9th column: 4*A006325(n+2); 11th column: -8*(1,8,34,104) (from columns 4,5,6,7 of F(n,m)).
As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - Philippe Deléham, Feb 21 2012

			Triangle T(n,k) begins:
  1
  2, 1
  1, 3,  1
  0, 4,  4,  1
  0, 4,  8,  5,   1
  0, 4, 12, 13,   6,   1
  0, 4, 16, 25,  19,   7,   1
  0, 4, 20, 41,  44,  26,   8,  1
  0, 4, 24, 61,  85,  70,  34,  9,  1
  0, 4, 28, 85, 146, 155, 104, 43, 10, 1
- _Philippe Deléham_, Feb 21 2012

Muniru A Asiru, Table of n, a(n) for n = 0..5151

Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).

Cf. A267633.

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)+Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)-Binomial(n-3,n-k-3)))); # Muniru A Asiru, Mar 22 2018

A130713 := proc(n)
    if n <= 2 and n >=0 then
        op(n+1,[1,2,1]) ;
    else
        0;
    end if;
end proc:
A202241 := proc(n,m)
    option remember;
    if n < 0 then
        0 ;
    elif m = 0 then
        A130713(n);
    else
        procname(n,m-1)+procname(n-1,m) ;
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A202241(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 22 2011
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
     seq(C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), k = 0..n);
end do; # Peter Bala, Mar 20 2018

rows = 12;
T[0] = PadRight[{1, 2, 1}, rows];
T[n_ /; nJean-François Alcover, Jun 29 2019 *)

def Trow(n): return [binomial(n, n-k) + binomial(n-1, n-k-1) - binomial(n-2, n-k-2) - binomial(n-3, n-k-3) for k in (0..n)]
for n in (0..9): print(Trow(n)) # Peter Luschny, Mar 21 2018

F(1,m) = m+2.
F(2,m) = A034856(m+1).
F(3,m) = A000297(m-1).
Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).
As a triangular array T(n,k), 0 <= k <= n, satisfies: T(n,k) = T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 2, T(2,0) = 1, T(3,0) = 0. - Philippe Deléham, Feb 21 2012
Unsigned diagonals of A267633 (beginning with its main diagonal) appear to be the reverse rows of this entry's triangle beginning with the fourth row. - Tom Copeland, Jan 26 2016
T(n,k) = C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), where C(n, k) = n!/(k!*(n-k)!) if 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

Original entry on oeis.org

0, 1, 0, 4, 3, 1, 9, 8, 6, 3, 16, 15, 13, 10, 6, 1, 25, 24, 22, 19, 15, 10, 4, 36, 35, 33, 30, 26, 21, 15, 8, 0, 49, 48, 46, 43, 39, 34, 28, 21, 13, 4, 64, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9, 81, 80, 78, 75, 71, 66, 60, 53, 45, 36, 26, 15, 3, 100, 99, 97

Offset: 0

Philippe Deléham, Mar 09 2013

Row lengths are in A214857.

			Triangle begins:
    0;
    1,   0;
    4,   3,   1;
    9,   8,   6,   3;
   16,  15,  13,  10,   6,   1;
   25,  24,  22,  19,  15,  10,   4;
   36,  35,  33,  30,  26,  21,  15,  8,  0;
   49,  48,  46,  43,  39,  34,  28, 21, 13,  4;
   64,  63,  61,  58,  54,  49,  43, 36, 28, 19,  9;
   81,  80,  78,  75,  71,  66,  60, 53, 45, 36, 26, 15,  3;
  100,  99,  97,  94,  90,  85,  79, 72, 64, 55, 45, 34, 22,  9;
  121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66, 55, 43, 30, 16, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. Diagonals: A000217, A034856, A055999,

Columns: A000290, A005563, A028872, A028878.

Table[s = {}; k = 0; While[tri = k*(k + 1)/2; tri <= n^2, AppendTo[s, n^2 - tri]; k++]; s, {n, 0, 10}] (* T. D. Noe, Mar 11 2013 *)

T(2*n,n) = A022264(n).

T(n,n) = n*(n-1)/2 = A000217(n-1).

A214928 A209293 as table read layer by layer clockwise.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 14, 7, 6, 8, 12, 17, 23, 20, 11, 10, 13, 19, 26, 34, 43, 30, 27, 16, 15, 18, 24, 31, 39, 48, 58, 53, 38, 35, 22, 21, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 32, 40, 49, 59, 70, 82, 95, 109, 102, 81, 76, 57, 54, 37, 36, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). The order of the list:
T(1,1)=1;
T(1,2), T(2,2), T(2,1);
. . .
T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);
. . .

			The start of the sequence as table:
  1....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,4,3;
  5,9,14,7,6;
  8,12,17,23,20,11,10;
  13,19,26,34,43,30,27,16,15;
  18,24,31,39,48,58,53,38,35,22,21;
  . . .
Row number r contains 2*r-1 numbers.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A209293, A209279, A209278, A185180, A060734, A060736; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=min(t,n-(t-1)**2)
j=min(t,t**2-n+1)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2), where i=min(t; n-(t-1)^2), j=min(t; t^2-n+1), t=floor(sqrt(n-1))+1.

A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

Original entry on oeis.org

1, 3, 4, 2, 5, 9, 14, 7, 6, 10, 11, 20, 23, 17, 12, 8, 13, 19, 26, 34, 43, 30, 27, 16, 15, 21, 22, 35, 38, 53, 58, 48, 39, 31, 24, 18, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 36, 37, 54, 57, 76, 81, 102, 109, 95, 82, 70, 59, 49, 40, 32, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 2013

nonn

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2);
. . .
T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);
T(2*m,1),   T(2*m,2),   ... T(2*m,2*m-1), T(2*m,2*m),     T(2*m-1,2*m), ... T(1,2*m);
. . .
The first row is layer read clockwise, the second row is layer counterclockwise.

			The start of the sequence as table:
  1....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  3,4,2;
  5,9,14,7,6;
  10,11,20,23,17,12,8;
  13,19,26,34,43,30,27,16,15;
  21,22,35,38,53,58,48,39,31,24,18;
  . . .
Row number r contains 2*r-1 numbers.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A081344, A209293, A209279, A209278, A185180; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where
m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2),
where i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.

A307213 Divide the natural numbers into sets of successive sizes 3,4,5,6,7,...,, starting with {1,2,3}. Cycle through each set until you reach a prime; if the prime was the n-th element in its set, jump to the n-th element of the next set.

Original entry on oeis.org

1, 2, 5, 9, 10, 11, 16, 17, 23, 30, 31, 39, 40, 41, 50, 51, 52, 43, 53, 64, 65, 66, 67, 79, 92, 93, 94, 95, 96, 97, 111, 112, 113, 128, 129, 130, 131, 147, 148, 149, 166, 167, 185, 186, 187, 169, 170, 171, 172, 173, 192, 193, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 244, 245, 246, 247, 248, 249, 250, 229

Offset: 1

Christopher Cormier, Mar 28 2019

nonn

"Cycle" in the definition, means that if no prime is found, go back to the start of the set.
If a set does not contain a prime, the sequence goes into an infinite loop, but it is conjectured that this does not happen since the sets are of increasing length.
The sets (rather intervals) are I_j = [(j^2 + 3*j - 2)/2, j*(j + 5)/2] =[A034856(j), A095998(j)], for j >= 1. For the  number of primes in these intervals see A309121. - Wolfdieter Lang, Jul 13 2019

			The first set is {1,2,3}. We look at 1 then 2. 2 is prime, and it is the second number in the set. The next set is {4,5,6,7}. So we jump to the second element, 5. 5 is also prime, so we jump to the second element of the next set, {8,9,10,11,12}, which is 9, etc. If we reach the end of a set without reaching a prime, we loop back to the first element, which is the only way for a(n) < a(n-1) to happen.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A307213

Cf. A034856, A095998, A309121.

Nest[Append[#1, {If[#3 <= Length@ #4, #3, #3 - Length@ #4], If[#2 == #3, {#4[[#3]]}, Join[#4, #4][[#2 ;; #3]]], #4}] & @@ {#1, #2, If[PrimeQ[#3[[#2]] ], #2, #2 + FirstPosition[RotateLeft[#3, #2], ?PrimeQ][[1]] ], #3} & @@ {#1, #2, Range[#3, #3 + #4]} & @@ {#, #[[-1, 1]], 1 + Max@ #[[-1, -1]], Length@ # + 2} &, {{#1, #2[[1 ;; #1]], #2} & @@ {FirstPosition[#, ?PrimeQ][[1]], #}} &@ Range@ 3, 19][[All, 2]] // Flatten (* Michael De Vlieger, Mar 31 2019 *)

PARI
```
See Links section.
```

Edited by N. J. A. Sloane, Jul 13 2019

A323220 a(n) = n(n + 5)(n + 7)*(n + 10)/24 + 1.

Original entry on oeis.org

1, 23, 64, 131, 232, 376, 573, 834, 1171, 1597, 2126, 2773, 3554, 4486, 5587, 6876, 8373, 10099, 12076, 14327, 16876, 19748, 22969, 26566, 30567, 35001, 39898, 45289, 51206, 57682, 64751, 72448, 80809, 89871, 99672, 110251, 121648, 133904, 147061, 161162, 176251

Offset: 0

Peter Luschny, Jan 25 2019

Cf. A323224 (column 5), A323233 (row 5), A323221 (first diff.), A034856 (second diff.).

a := n -> (n^4 + 22*n^3 + 155*n^2 + 350*n + 24)/24:
seq(a(n), n=0..40);

a(n) = [x^n] (8*x^4 - 31*x^3 + 41*x^2 - 18*x - 1)/(x - 1)^5.
a(n) = n! [x^n] exp(x)*(x^4 + 28*x^3 + 228*x^2 + 528*x + 24)/24.
a(n) = (1/3)*((2*n + 17)*a(n-3) - (3*n + 25)*a(n-2) + (n + 15)*a(n-1)) for n >= 3.
a(n) = A323224(n, 5).

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A323220 a(n) = n(n + 5)(n + 7)*(n + 10)/24 + 1.