A104677 -id:A104677 - cp's OEIS frontend

A062190 Coefficient triangle of certain polynomials N(5; m,x).

1, 1, 6, 1, 14, 21, 1, 24, 84, 56, 1, 36, 216, 336, 126, 1, 50, 450, 1200, 1050, 252, 1, 66, 825, 3300, 4950, 2772, 462, 1, 84, 1386, 7700, 17325, 16632, 6468, 792, 1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287, 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002

Offset: 0

Wolfdieter Lang, Jun 19 2001

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.

			Triangle begins as:
  1;
  1,   6;
  1,  14,   21;
  1,  24,   84,    56;
  1,  36,  216,   336,    126;
  1,  50,  450,  1200,   1050,    252;
  1,  66,  825,  3300,   4950,   2772,     462;
  1,  84, 1386,  7700,  17325,  16632,    6468,    792;
  1, 104, 2184, 16016,  50050,  72072,   48048,  13728,   1287;
  1, 126, 3276, 30576, 126126, 252252,  252252, 123552,  27027,  2002;
  1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, 3003;

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

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
Diagonals: A000389 (k=n), A027818 (k=n-1), A104670 (k=n-2), A104671 (k=n-3), A104672 (k=n-4), A104673 (k=n-5), A104674 (k=n-6).
Cf. A003516 (row sums), A113894 (main diagonal).

A062190:= func< n,k | Binomial(n,k)*Binomial(n+5,k) >;
[A062190(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2025

A062190 := proc(m,k)
    add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Nov 29 2015

NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
A062190[n_,k_]:= Binomial[n,k]*Binomial[n+5,k];
Table[A062190[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2025 *)

def A062190(n,k): return binomial(n,k)*binomial(n+5,k)
print(flatten([[A062190(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 28 2025

T(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). - Jean-François Alcover, Sep 18 2013
T(n, k) = binomial(n, k)*binomial(n+5, k). - G. C. Greubel, Feb 28 2025

A265010 Numbers which are the product of two tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 10, 16, 20, 35, 40, 56, 80, 84, 100, 120, 140, 165, 200, 220, 224, 286, 336, 350, 364, 400, 455, 480, 560, 660, 680, 700, 816, 840, 880, 969, 1120, 1140, 1144, 1200, 1225, 1330, 1456, 1540, 1650, 1680, 1771, 1820, 1960, 2024, 2200, 2240

Offset: 1

R. J. Mathar, Nov 30 2015

nonn

This is for the tetrahedral numbers A000292 what A085780 is for the triangular numbers.

The subsequence of numbers with more than one factorization starts 0, 560 (= 65*10 = 1*560), 19600 (= 560*15 = 19600*1), 28560 (=816 *35 = 7140*4), 43680, 292600, 416640, ...

			Contains 480=4*120, 560=1*560, 660=4*165, 680=1*680, 700=20*35, ....

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A085780. Subsequences: A000292, A001249, A027790, A105939, A104474, A104677.

# reuses code of A000292
isA265010 := proc(n)
    if n = 0 then
        return true;
    end if;
    for d in numtheory[divisors](n) do
        if isA000292(d) and isA000292(n/d) then
            return true;
        end if;
    end do:
    false;
end proc:
for n from 0 to 4000 do
    if isA265010(n) then
        printf("%d, ",n);
    end if;
end do:

lim = 2240; t = Table[Binomial[n + 2, 3], {n, 0, 10^3}]; f[n_] := Select[{#, n/#} & /@ Select[Divisors[n], # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # > 0 &] (* Michael De Vlieger, Nov 30 2015 *)

{n: n = A000292(i)*A000292(j) for some i,j>=0}.

cp's OEIS Frontend

A062190 Coefficient triangle of certain polynomials N(5; m,x).

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