A128503 -id:A128503 - cp's OEIS frontend

A128504 Row sums of array A128503 (second convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials).

1, 3, 3, -2, -9, -9, 3, 18, 18, -4, -30, -30, 5, 45, 45, -6, -63, -63, 7, 84, 84, -8, -108, -108, 9, 135, 135, -10, -165, -165, 11, 198, 198, -12, -234, -234, 13, 273, 273, -14, -315, -315, 15, 360, 360, -16, -408, -408, 17, 459, 459

Offset: 0

Wolfdieter Lang Apr 04 2007

Second convolution of A010892.
Convolution of A099254 with A010892.
a(n) equals the coefficient of x^2 of the characteristic polynomial of the (n+2)X(n+2) tridiagonal matrix with 1's along the main diagonal, the superdiagonal, and the subdiagonal (see Mathematica code below). [John M. Campbell, Jul 10 2011]

Tewodros Amdeberhan, George E. Andrews, and Roberto Tauraso, Further study on MacMahon-type sums of divisors, arXiv:2409.20400 [math.NT], 2024. See p. 18.

Cf. A010892, A099254, A128503.

Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 - 1] + KroneckerDelta[#1, #2 + 1] &, {n + 2, n + 2}], x], x^2], {n, 0, 70}] (* John M. Campbell, Jul 10 2011 *)

Vec(1/(1-x+x^2)^3+O(x^66)) \\ Joerg Arndt, Jul 02 2013

a(n) = Sum_{m=0..floor(n/2)} A128503(n,m).
G.f.: 1/(1-x+x^2)^3.
a(n) = (floor(n/3)+1)*(floor(n/3)-floor((n-1)/3)+(3/2)*(floor(n/3)+2)*(3*floor((n+1)/3)-n))*(-1)^n. - Tani Akinari, Jul 03 2013

A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 12, 6, 0, 0, 10, 30, 30, 10, 0, 0, 15, 60, 90, 60, 15, 0, 0, 21, 105, 210, 210, 105, 21, 0, 0, 28, 168, 420, 560, 420, 168, 28, 0, 0, 36, 252, 756, 1260, 1260, 756, 252, 36, 0, 0, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 0, 0, 55, 495

Offset: 0

Emeric Deutsch, Apr 22 2007

Previous name was: Triangle read by rows: T(n,k)=derivative of the q-binomial coefficient [n,k] evaluated at q=1 (0<=k<=n). - N. J. A. Sloane, Jan 06 2016
For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1. - Emeric Deutsch, Apr 22 2007
Sum of entries in n-th antidiagonal = n(n-1)2^(n-3) = A001788(n-1).
T(n,k) = A094305(n-2, k-1) for n >= 2, k >= 1.
T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered n-tuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <=  ... <= i_n <= k. - Alan Shore and N. J. A. Sloane, Jan 06 2016
T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n-1, k-1, 2). - N. J. A. Sloane, Jan 06 2016
As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,n-k). - Geoffrey Critzer, Mar 22 2018

			Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... (A000004)
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... (A000217)
0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, ... (A027480)
0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, ... (A033487)
0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, ... (A266732)
0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, ... (A240440)
0, 21, 168, 756, 2520, 6930, 16632, 36036, ... (A266733)
...
If regarded as a triangle, this begins:
  0;
  0,  0;
  0,  1,  0;
  0,  3,  3,  0;
  0,  6, 12,  6,  0;
  0, 10, 30, 30, 10,  0;
  0, 15, 60, 90, 60, 15, 0;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

Cemil Karaçam and Alper Vural, Enumerating 2D and 3D lattice paths with arbitrary steps, Mathematica Bohemica, pp. 1-13 (2025). See p. 5.
Index entries for sequences related to dominoes.

Cf. A001788.
Rows give A000004, A000217, A027480, A033487, A266732, A240440, A266733.
A128503 and A094305 are very similar sequences.

dd:=proc(n,m) if m=0 or n=0 then 0 else (m+n)!/(2*(m-1)!*(n-1)!); fi; end;
f:=n->[seq(dd(n,m),m=0..30)];
for n from 0 to 10 do lprint(f(n)); od: # produces sequence as square array
T:=(n,k)->k*(k+1)*binomial(n,k+1)/2: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[Table[D[Expand[FunctionExpand[QBinomial[n, k, q]]], q] /. q -> 1, {k, 0, n}], {n, 0, 15}] // Grid (* Geoffrey Critzer, Mar 22 2018 *)

T(n,k) = (1/2)*k*(k+1)*binomial(n,k+1).

G.f.: G(q,z) = qz^2/(1-z-qz)^3.

Entry revised by N. J. A. Sloane, Jan 06 2016

A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

Original entry on oeis.org

1, 4, 10, -4, 20, -20, 35, -60, 10, 56, -140, 60, 84, -280, 210, -20, 120, -504, 560, -140, 165, -840, 1260, -560, 35, 220, -1320, 2520, -1680, 280, 286, -1980, 4620, -4200, 1260, -56, 364, -2860, 7920, -9240, 4200, -504, 455, -4004, 12870, -18480, 11550, -2520, 84, 560, -5460, 20020, -34320

Offset: 0

Wolfdieter Lang, Apr 04 2007

S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m) with the second convolution S2(n,x) given by array A128503.

Row polynomials P3(n,x) := Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x).

			  1;
  4;
  10,   -4;
  20,  -20;
  35,  -60,  10;
  56, -140,  60;
  84, -280, 210,  -20;
  120,-504, 560, -140;
  ...
n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.

Wolfdieter Lang, First 15 rows and more.

Row sums (signed array) give A128506. Unsigned row sums are A001872.

Cf. A128503 (k=2 convolution).

a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0.
a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0.
G.f. for S3(n,x): 1/(1-x*z+z^2)^4.
G.f. for P3(n,x): 1/(1-z+x*z^2)^4.

Name edited by Petros Hadjicostas, Sep 04 2019

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A128504 Row sums of array A128503 (second convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials).

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A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

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A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

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