A382614 -id:A382614 - cp's OEIS frontend

A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 47, 8, 0, 1, 6, 40, 124, 198, 118, 13, 0, 1, 7, 57, 235, 571, 681, 290, 21, 0, 1, 8, 77, 398, 1320, 2500, 2263, 702, 34, 0, 1, 9, 100, 623, 2640, 7026, 10504, 7341, 1677, 55, 0

Offset: 0

Seiichi Manyama, Apr 08 2023

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   7,  15,   26,   40, ...
  0, 3,  18,  55,  124,  235, ...
  0, 5,  47, 198,  571, 1320, ...
  0, 8, 118, 681, 2500, 7026, ...

Columns k=0..3 give A000007, A000045(n+1), A362126, A382614.
Main diagonal gives A362080.
Cf. A362078, A362079.

T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(k*j, n-j));

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(k*j,n-j).

A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

Original entry on oeis.org

1, 3, 12, 37, 111, 315, 864, 2307, 6027, 15471, 39132, 97755, 241606, 591636, 1437078, 3465748, 8305161, 19788957, 46910232, 110686101, 260064912, 608684490, 1419591546, 3300027546, 7648265728, 17676484410, 40747630332, 93704299336, 214999206831, 492262973433

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (3,3,-8,-12,3,17,15,6,1).

Cf. A002478, A362126.

Cf. A000217, A001628, A382614.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^2)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, n-k));

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,n-k).
a(n) = 3*a(n-1) + 3*a(n-2) - 8*a(n-3) - 12*a(n-4) + 3*a(n-5) + 17*a(n-6) + 15*a(n-7) + 6*a(n-8) + a(n-9).
G.f.: 1/(1-x-2*x^2-x^3)^3. - Vincenzo Librandi, Apr 10 2025

A382613 Expansion of 1/(1 - x*(1 + x)^3)^2.

Original entry on oeis.org

1, 2, 9, 28, 88, 270, 808, 2386, 6960, 20104, 57607, 163950, 463907, 1306104, 3661248, 10223820, 28452400, 78941412, 218426608, 602886704, 1660329597, 4563175466, 12517834605, 34280427828, 93729509848, 255900484218, 697712467704, 1899912606358, 5167488465184

Offset: 0

Seiichi Manyama, Mar 31 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,5,0,-13,-20,-15,-6,-1).

Cf. A099234, A382614.

Cf. A001629, A362126.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 08 2025

Table[Sum[(k+1)*Binomial[3*k,n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, Apr 08 2025 *)

a(n) = sum(k=0, n, (k+1)*binomial(3*k, n-k));

a(n) = Sum_{k=0..n} (k+1) * binomial(3*k,n-k).
a(n) = 2*a(n-1) + 5*a(n-2) - 13*a(n-4) - 20*a(n-5) - 15*a(n-6) - 6*a(n-7) - a(n-8).
G.f.: 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2. - Vincenzo Librandi, Apr 08 2025

A382892 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^3.

Original entry on oeis.org

1, 3, 24, 190, 1659, 15309, 146986, 1453536, 14704917, 151479031, 1583533308, 16756882194, 179149227231, 1932144798513, 20996553430206, 229678298803028, 2527034248221849, 27947027713469307, 310494250880357488, 3463870813896354726, 38787008808135775299

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A360076, A382894.

Cf. A366272, A382614.

a(n, r=3, s=3, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x)^3 * A(x)^(4/3) )^3.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A366272.

cp's OEIS Frontend

A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

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A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

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A382613 Expansion of 1/(1 - x*(1 + x)^3)^2.

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A382892 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^3.

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