A027803 -id:A027803 - cp's OEIS frontend

A053347 a(n) = binomial(n+7, 7)*(n+4)/4.

1, 10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068, 119340, 201552, 329460, 523260, 810084, 1225785, 1817046, 2643850, 3782350, 5328180, 7400250, 10145070, 13741650, 18407025, 24402456, 32040360, 41692024, 53796160

Offset: 0

Barry E. Williams, Jan 06 2000

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-9) is the number of 9-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
8-dimensional square numbers, seventh partial sums of binomial transform of [1, 2, 0, 0, 0, ...]. a(n) = sum{i=0,n,C(n+7, i+7)*b(i)}, where b(i) = [1, 2, 0, 0, 0, ...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 7 queens on an (n+7) X (n+7) chessboard so that they diagonally attack each other exactly 21 times. The maximal possible attack number, p=binomial(k,2)=21 for k=7 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015
Coefficients in the terminating series identity 1 - 10*n/(n + 9) + 54*n*(n - 1)/((n + 9)*(n + 10)) - 210*n*(n - 1)*(n - 2)/((n + 9)*(n + 10)*(n + 11)) + ... = 0 for n = 1,2,3,.... Cf. A050486. - Peter Bala, Feb 18 2019

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Index entries for sequences related to Chebyshev polynomials.

Partial sums of A050486.

Cf. A005585, A040977.

[Binomial(n+7,7)+2*Binomial(n+7,8): n in [0..35]]; // Vincenzo Librandi, Jun 09 2013

A053347:=n->binomial(n+7,7)*(n+4)/4: seq(A053347(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

s1=s2=s3=s4=s5=s6=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; s4+=s3; s5+=s4; s6+=s5; AppendTo[lst,s6],{n,0,7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
CoefficientList[Series[(1 + x) / (1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Table[SeriesCoefficient[(1 + x)/(1 - x)^9, {x, 0, n}], {n, 0, 28}] (* or *)
Table[Binomial[n + 7, 7] (n + 4)/4, {n, 0, 28}] (* Michael De Vlieger, Dec 31 2015 *)

a(n)=binomial(n+7,7)*(n+4)/4 \\ Charles R Greathouse IV, Jun 10 2011

A053347_list, m = [], [2]+[1]*8
for _ in range(10**2):
    A053347_list.append(m[-1])
    print(m[-1])
    for i in range(8):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

a(n) = ((-1)^n)*A053120(2*n+8, 8)/2^7 (1/128 of ninth unsigned column of Chebyshev T-triangle, zeros omitted).
G.f.: (1+x)/(1-x)^9.
a(n) = 2*C(n+8, 8) - C(n+7, 7). - Paul Barry, Mar 04 2003
a(n) = A027803(n-3)/35 = C(n+4, n)*C(n+7, 4)/35. - Zerinvary Lajos, May 25 2005
a(n) = C(n+7, 7) + 2*C(n+7, 8). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (n^8 + 32*n^7 + 434*n^6 + 3248*n^5 + 14609*n^4 + 40208*n^3 + 65596*n^2 + 57312*n + 20160)/20160. - Chai Wah Wu, Jan 24 2016
Sum_{n>=0} 1/a(n) = 41503/45 - 280/3*Pi^2. - Jaume Oliver Lafont, Jul 17 2017
Sum_{n>=0} (-1)^n/a(n) = 140*Pi^2/3 - 1379/3. - Amiram Eldar, Jan 25 2022

A062145 Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.

Original entry on oeis.org

1, 1, 4, 1, 10, 10, 1, 18, 45, 20, 1, 28, 126, 140, 35, 1, 40, 280, 560, 350, 56, 1, 54, 540, 1680, 1890, 756, 84, 1, 70, 945, 4200, 7350, 5292, 1470, 120, 1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165, 1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220

Offset: 0

Wolfdieter Lang, Jun 19 2001

Coefficient triangle of certain polynomials N(3; m,x).

			As a square array:
    1,    1,     1,     1,     1,     1,    1,  1, ... A000012;
    4,   10,    18,    28,    40,    54,   70, 88, ... A028552;
   10,   45,   126,   280,   540,   945, 1540, ....... A105938;
   20,  140,   560,  1680,  4200,  9240, ............. A105939;
   35,  350,  1890,  7350, 23100, 62370, ............. A027803;
   56,  756,  5292, 25872, 99792, .................... A105940;
   84, 1470, 12936, 77616, ........................... A105942;
  120, 2640, 28512, .................................. A105943;
  165, 4455, 57015, .................................. A105944;
  ....;
As a triangle:
  1;
  1,   4;
  1,  10,   10;
  1,  18,   45,    20;
  1,  28,  126,   140,    35;
  1,  40,  280,   560,   350,    56;
  1,  54,  540,  1680,  1890,   756,    84;
  1,  70,  945,  4200,  7350,  5292,  1470,   120;
  1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
  1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;
  ....;

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

Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
Columns: A028552 (k=1), A105938 (k=2), A105939 (k=3), A027803 (k=4), A105940 (k=5), A105942 (k=6), A105943 (k=7), A105944 (k=8).
Diagonals: A000292 (k=n), A027800 (k=n-1), A107417 (k=n-2), A107418 (k=n-3), A107419 (k=n-4), A107420 (k=n-5), A107421 (k=n-6), A107422 (k=n-7).
Sums: A002054 (row).
Cf. A000108, A000292, A047074.

A062145:= func< n,k | Binomial(n,k)*Binomial(n+3,k) >;
[A062145(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2025

NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/( (m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n-k+c, k] z^k /k!, {k,0,n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
CL[4]  (* Peter Luschny, Feb 12 2024 *)
A062145[n_,k_]:= Binomial[n,k]*Binomial[n+3,k];
Table[A062145[n,k], {n,0,12},{k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2025 *)

def A062145(n,k): return binomial(n,k)*binomial(n+3,k)
print(flatten([[A062145(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 07 2025

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). - Jean-François Alcover, Sep 18 2013
From G. C. Greubel, Mar 07 2025 : (Start)
T(n, k) = binomial(n, k)*binomial(n+3, k).
T(2*n, n) = (1/2)*(n+1)^2*A000108(n)*A000108(n+2).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^floor((n+2)/2)*(A047074(n+3) - A047074(n+ 2)). (End)

New name by Peter Luschny, Feb 12 2024

More terms from G. C. Greubel, Mar 07 2025

cp's OEIS Frontend

A053347 a(n) = binomial(n+7, 7)*(n+4)/4.

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