A056108 -id:A056108 - cp's OEIS frontend

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

1, 3, 1, 7, 5, 1, 13, 11, 7, 1, 21, 19, 15, 9, 1, 31, 29, 25, 19, 11, 1, 43, 41, 37, 31, 23, 13, 1, 57, 55, 51, 45, 37, 27, 15, 1, 73, 71, 67, 61, 53, 43, 31, 17, 1, 91, 89, 85, 79, 71, 61, 49, 35, 19, 1, 111, 109, 105, 99, 91, 81, 69, 55, 39, 21, 1, 133, 131, 127, 121, 113, 103, 91, 77, 61, 43, 23, 1, 157, 155, 151, 145, 137, 127, 115, 101, 85, 67, 47, 25, 1, 183, 181, 177, 171, 163, 153, 141, 127, 111, 93, 73, 51, 27, 1

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).

			Northwest corner:
  1, 3,  7, 13, 21
  1, 5, 11, 19, 29
  1, 7, 15, 25, 45
  1, 9, 19, 31, 45

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185878, A185879, A185880.
Row 1 to 3: A002061, A028387, A082111.
diag (1,5,...): A056108;
diag (3,11,...): A056106;
diag (7,19,...): A003215;
diag (13,29,...): A144391;
diag (1,7,...): A003215;
diag (1,9,...): A144390.

(* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=k^2+(2n-3)k-2n+3;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185877 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185878 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185879 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.

A185880 Second accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 5, 3, 16, 17, 6, 40, 56, 38, 10, 85, 140, 128, 70, 15, 161, 295, 320, 240, 115, 21, 280, 553, 670, 600, 400, 175, 28, 456, 952, 1246, 1250, 1000, 616, 252, 36, 705, 1536, 2128, 2310, 2075, 1540, 896, 348, 45, 1045, 2355, 3408, 3920, 3815, 3185, 2240, 1248, 465, 55, 1496, 3465, 5190, 6240, 6440, 5831, 4620, 3120, 1680, 605, 66, 2080, 4928, 7590, 9450, 10200, 9800, 8428, 6420, 4200, 2200

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.

			Northwest corner:
   1,    5,   16,   40,   85
   3,   17,   56,  140,  295
   6,   38,  128,  320,  670
  10,   70,  240,  600, 1250

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185878.
Antidiagonal sums: A037235.
diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
diag (7,19,...): A003215 (hex numbers);
diag (13,29,...): A144391.

(* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.

A247612 a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).

Original entry on oeis.org

1, 15, 120, 680, 3060, 11628, 38760, 116280, 316767, 788161, 1805100, 3840420, 7660250, 14446134, 25947612, 44668692, 74091645, 118941555, 185495056, 281936688, 418766304, 609260960, 869994720, 1221419808, 1688512539, 2301487461, 3096583140, 4116923020

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A056108, A247608 - A247611

[(1680+386012*n-958048*n^2+943761*n^3-455455*n^4+123123*n^5- 17017*n^6+1144*n^7)/1680: n in [0..40]];

I:=[1,15,120,680,3060,11628,38760, 116280]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]];

Table[(1680 + 386012 n - 958048 n^2 + 943761 n^3 - 455455 n^4 + 123123 n^5 - 17017 n^6 + 1144 n^7)/1680, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 7 x + 28 x^2 + 84 x^3 + 210 x^4 + 462 x^5 + 924 x^6 + 1716 x^7)/(1 - x)^8, {x, 0, 40}], x]
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,15,120,680,3060,11628,38760,116280},30] (* Harvey P. Dale, May 12 2017 *)

m=7; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 7*x + 28*x^2 + 84*x^3 + 210*x^4 + 462*x^5 + 924*x^6 + 1716*x^7) / (1-x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = (1680 + 386012*n - 958048*n^2 + 943761*n^3 - 455455*n^4 + 123123*n^5 - 17017*n^6 + 1144*n^7)/1680.

Definition edited by Robert Israel, Sep 22 2014

A113653 Isolated semiprimes in the hexagonal spiral.

Original entry on oeis.org

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.
Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.
An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].
Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - N. J. A. Sloane, Jun 27 2020

			The spiral begins:
                120-119-118-117-116-115-114
                 /                         \
              121  85--84--83-*82*-81--80 113
               /   /                     \   \
            122  86  56--55--54--53--52  79 112
             /   /   /                 \   \   \
          123  87  57  33--32--31--30 *51* 78 111
           /   /   /   /             \   \   \   \
        124  88  58  34  16--15--14  29  50  77 110
         /   /   /   /   /         \   \   \   \   \
      125  89  59  35  17   5---4  13  28  49  76 109
       /   /   /   /   /   /     \   \   \   \   \   \
    126  90  60  36  18  *6*  0   3  12  27  48  75 108
     /   /   /   /   /   /   /   /   /   /   /   /   /
  127 *91* 61  37  19   7   1---2  11  26  47  74 107 146
     \   \   \   \   \   \         /   /   /   /   /   /
    128  92  62  38  20   8---9--10  25  46  73 106 145
       \   \   \   \   \             /   /   /   /   /
      129  93  63  39  21--22--23--24  45  72 105 144
         \   \   \   \                 /   /   /   /
        130  94  64  40--41--42--43--44  71 104 143
           \   \   \                     /   /   /
          131  95  65--66--67--68-*69*-70 103 142
             \   \                         /   /
            132  96--97--98--99-100-101-102 141
               \                             /
              133-134-135-136-137-138-139-140

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
R. J. Mathar, Maple program for A113653
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A113688.
For the sequence of isolated primes see A335916.
Related sequences:
A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).
A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).
A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).
A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).
A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).
A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

Corrected and edited by N. J. A. Sloane, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

A134234 Triangle read by rows, n-th row = n terms of 2n, 2n+2, 2n+4, ..., 1; with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 6, 8, 1, 8, 10, 12, 1, 10, 12, 14, 16, 1, 12, 14, 16, 18, 20, 1, 14, 16, 18, 20, 22, 24, 1, 16, 18, 20, 22, 24, 26, 28, 1, 18, 20, 22, 24, 26, 28, 30, 32, 1, 20, 22, 24, 26, 28, 30, 32, 34, 36, 1

Offset: 1

Gary W. Adamson, Oct 14 2007

Row sums = A056108: (1, 5, 15, 31, 53, ...).

			First few rows of the triangle:
   1;
   4,  1;
   6,  8,  1;
   8, 10, 12,  1;
  10, 12, 14, 16, 1;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011

Cf. A056108.

seq(op([seq(2*n+2*k,k=0..n-2),1]),n=1..10); # Robert Israel, Jan 15 2016

Flatten[Table[Join[{1},Range[n,2n-3,2]],{n,4,30,2}]] (* Harvey P. Dale, Nov 06 2013 *)

G.f.: 2*x/(1-x)^2 - 2*(2-x)/(1-x)*Sum_{n>=1} n*x^(n*(n+1)/2) + (3-x)/(1-x)*Sum_{n>=1} x^(n*(n+1)/2). - Robert Israel, Jan 15 2016

A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

Original entry on oeis.org

1, 9, 45, 165, 495, 1231, 2639, 5055, 8885, 14605, 22761, 33969, 48915, 68355, 93115, 124091, 162249, 208625, 264325, 330525, 408471, 499479, 604935, 726295, 865085, 1022901, 1201409, 1402345, 1627515, 1878795, 2158131, 2467539, 2809105, 3184985, 3597405

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005408, A056108, A247608.

[(12-58*n+217*n^2-98*n^3+35*n^4)/12: n in [0..40]];

[1+8*Binomial(n, 1)+28*Binomial(n, 2)+56*Binomial(n, 3)+70*Binomial(n,4): n in [0..40]];

I:=[1,9,45,165,495]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[(12 - 58 n + 217 n^2 - 98 n^3 + 35 n^4)/12, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 4 x + 10 x^2 + 20 x^3 + 35 x^4)/(1 - x)^5, {x, 0, 50}], x]
LinearRecurrence[{5,-10,10,-5,1},{1,9,45,165,495},40] (* Harvey P. Dale, Oct 19 2024 *)

m=4; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (12 - 58*n + 217*n^2 - 98*n^3 + 35*n^4)/12.
a(n) = 1 + 8*Binomial(n, 1) + 28*Binomial(n, 2) + 56*Binomial(n, 3) + 70*Binomial(n, 4).

A247611 a(n) = Sum_{k=0..6} binomial(12,k)*binomial(n,k).

Original entry on oeis.org

1, 13, 91, 455, 1820, 6188, 18564, 49596, 119139, 260743, 527065, 996205, 1778966, 3027038, 4942106, 7785882, 11891061, 17673201, 25643527, 36422659, 50755264, 69525632, 93774176, 124714856, 163753527, 212507211, 272824293, 346805641, 436826650

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A005408, A056108, A247608, A247609.

[(120-8042*n+20581*n^2-17380*n^3+7645*n^4-1518*n^5+ 154*n^6)/120: n in [0..40]];

I:=[1,13,91,455,1820,6188,18564]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]];

Table[(120 - 8042 n + 20581 n^2 - 17380 n^3 + 7645 n^4 - 1518 n^5 + 154 n^6)/120, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 6 x + 21 x^2 + 56 x^3 + 126 x^4 + 252 x^5 + 462 x^6)/(1 - x)^7, {x, 0, 40}], x]

m=6; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 6*x + 21*x^2 + 56*x^3 + 126*x^4 + 252*x^5 + 462*x^6) / (1-x)^7.
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).
a(n) = (120 - 8042*n + 20581*n^2 - 17380*n^3 + 7645*n^4 -1518*n^5 + 154*n^6)/120.

A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 19, 15, 6, 1, 1, 7, 21, 31, 31, 21, 7, 1, 1, 8, 28, 46, 53, 46, 28, 8, 1, 1, 9, 36, 64, 81, 81, 64, 36, 9, 1, 1, 10, 45, 85, 115, 126, 115, 85, 45, 10, 1, 1, 11, 55, 109, 155, 181, 181, 155, 109, 55, 11, 1

Offset: 0

Kolosov Petro, Jul 06 2024

Triangle T(n,k) is the second triangle R2 among the rascal-family triangles; A374452 is triangle R3; A077028 is triangle R1.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 5).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 2).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  19   15    6    1
n=7:   1   7  21  31   31   21    7   1
n=8:   1   8  28  46   53   46   28   8   1
n=9:   1   9  36  64   81   81   64  36   9   1
n=10:  1  10  45  85  115  126  115  85  45  10  1

Kolosov Petro, Table of n, a(n) for n = 0..1325
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Jena Gregory, Brandt Kronholm, and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A007318, A000027, A000217, A005448, A056108, A212656, A000292, A095667, A095666, A006261.

t[n_, k_]:=Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 2}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Center]

T(n,k) = 1 + k*(n-k) + (1/4)*(k-1)*k*(n-k-1)*(n-k).
Row sums give A006261(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A005448(n).
Diagonal T(n+4, n) gives A056108(n).
Diagonal T(n+5, n) gives A212656(n).
Column k=3 difference binomial(n+6, 3) - T(n+6, 3) gives C(n+3,3)=A007318(n+3,3).
Column k=4 difference binomial(n+7, 4) - T(n+7, 4) gives fifth column of (1,4)-Pascal triangle A095667.
G.f.: (1 + 3*x^4*y^2 - (2*x + 3*x^3*y)*(1 + y) + x^2*(1 + 5*y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 09 2024

A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).

Original entry on oeis.org

1, 11, 66, 286, 1001, 3003, 7798, 17858, 36873, 70003, 124130, 208110, 333025, 512435, 762630, 1102882, 1555697, 2147067, 2906722, 3868382, 5070009, 6554059, 8367734, 10563234, 13198009, 16335011, 20042946, 24396526, 29476721, 35371011, 42173638, 49985858

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005408, A056108, A247608.

[(20+508*n-925*n^2+820*n^3-245*n^4+42*n^5)/20: n in [0..40]];

I:=[1,11,66,286,1001,3003]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]];

Table[(20 + 508 n - 925 n^2 + 820 n^3 - 245 n^4 + 42 n^5)/20, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 5 x + 15 x^2 + 35 x^3 + 70 x^4 + 126 x^5)/(1 - x)^6, {x, 0, 40}], x]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,66,286,1001,3003},40] (* Harvey P. Dale, Apr 20 2022 *)

m=5; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 126*x^5) / (1-x)^6.
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).
a(n) = (20 + 508*n - 925*n^2 + 820*n^3 - 245*n^4 + 42*n^5)/20.

A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

Original entry on oeis.org

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2031535, 5189327, 12316239, 27322191, 57029103, 112740255, 212383935, 383358645, 666220005, 1119362365, 1824861005, 2895653673, 4484253081, 6793194849, 10087438257, 14708950035, 21093714291

Offset: 0

Vincenzo Librandi, Sep 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A005408, A056108, A247608 - A247612.

m:=8; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];

[(20160-15076944*n+40499716*n^2-42247940*n^3 +23174515*n^4-7234136*n^5+1335334*n^6-134420*n^7 +6435*n^8)/20160: n in [0..40]];

Table[(20160 - 15076944 n + 40499716 n^2 - 42247940 n^3 + 23174515 n^4 - 7234136 n^5 + 1335334 n^6 - 134420 n^7 + 6435 n^8)/20160, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 8 x + 36 x^2 + 120 x^3 + 330 x^4 + 792 x^5 + 1716 x^6 + 3432 x^7 + 6435 x^8)/(1 - x)^9, {x, 0, 40}], x]
Table[Sum[Binomial[16,k]Binomial[n,k],{k,0,8}],{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,153,969,4845,20349,74613,245157,735471},40] (* Harvey P. Dale, Mar 25 2015 *)

m=8; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

G.f.: (1 + 8*x + 36*x^2 + 120*x^3 + 330*x^4 + 792*x^5 + 1716*x^6 + 3432*x^7  + 6435*x^8) / (1-x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
a(n) = (20160 - 15076944*n + 40499716*n^2 - 42247940*n^3 + 23174515*n^4 - 7234136*n^5 + 1335334*n^6 - 134420*n^7 + 6435*n^8) / 20160.

cp's OEIS Frontend

A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

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A185880 Second accumulation array of A185877, by antidiagonals.

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A247612 a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).

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A113653 Isolated semiprimes in the hexagonal spiral.

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A134234 Triangle read by rows, n-th row = n terms of 2n, 2n+2, 2n+4, ..., 1; with a(1) = 1.

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A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

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A247611 a(n) = Sum_{k=0..6} binomial(12,k)*binomial(n,k).

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A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.