A002663 -id:A002663 - cp's OEIS frontend

A368535 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^(n-k).

0, 1, 6, 31, 188, 1510, 16106, 217938, 3577624, 68952495, 1524157870, 37974983321, 1052320304212, 32089921353308, 1067586804710258, 38470738234990580, 1492501011869912496, 62015249735222969325, 2747431806313734355830, 129267455591507496073315

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A062805, A368534.

Cf. A002663, A097786, A097788.

a(n) = sum(k=1, n, binomial(k+2, 3)*n^(n-k));

a(n) = [x^n] x/((1-n*x) * (1-x)^4).

a(n) = n * (6*n^(n+2) - n^5 - 3*n^4 + n^3 + n^2 - 6*n + 2)/(6 * (n-1)^4) for n > 1.

A229702 Expansion of 1/((1-x)^4*(1-6x)).

Original entry on oeis.org

1, 10, 70, 440, 2675, 16106, 96720, 580440, 3482805, 20897050, 125382586, 752295880, 4513775735, 27082654970, 162495930500, 974975583816, 5849853503865, 35099121024330, 210594726147310, 1263568356885400, 7581410141314171

Offset: 0

Yahia Kahloune, Sep 27 2013

This sequence was chosen to illustrate a way to match generating functions and closed-form solutions.
The general term associated with the generating function
1/((1-s*x)^4*(1-r*x)) with r>s>=1 is a(n) = [ r^(n+4) - s^(n+1)*(s^3 + s^2*(r-s)*binomial(n+4,1) + s*(r-s)^2*binomial(n+4,2)+(r-s)^3*binomial(n+4,3))]/(r-s)^4.

			a(3) = (6^8 - (125*3^3  + 1200*3^2 + 3805*3 + 4026))/3750 = 440.

Index entries for linear recurrences with constant coefficients, signature (10,-30,40,-25,6).

Cf. A002663, A097786, A097788, A097790.

a(n) = (6^(n+4) - (1 + 5*C(n+4,1) + 25*C(n+4,2) + 125*C(n+4,3)))/625 = (6^(n+5) - (125*n^3 + 1200*n^2 + 3805*n + 4026))/3750.

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

Original entry on oeis.org

2, 3, -2, 4, -5, 2, 5, -9, 7, -2, 6, -14, 16, -9, 2, 7, -20, 30, -25, 11, -2, 8, -27, 50, -55, 36, -13, 2, 9, -35, 77, -105, 91, -49, 15, -2, 10, -44, 112, -182, 196, -140, 64, -17, 2, 11, -54, 156, -294, 378, -336, 204, -81, 19, -2

Offset: 1

Clark Kimberling, Sep 24 2022

n-th row sum = 1 for n >= 2.

			First 7 rows:
  2
  3      -2
  4      -5       2
  5      -9       7     -2
  6     -14      16     -9     2
  7     -20      30    -25    11     -2
  8     -27      50    -55    36    -13     2
Row 4 gives recurrence coefficients for the sequence
(r(k)) = (A002662(k)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
(Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)

Cf. A029638, A029635.

Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.

Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]

T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.

T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.

cp's OEIS Frontend

A368535 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^(n-k).

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A229702 Expansion of 1/((1-x)^4*(1-6x)).

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

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