A071816 -id:A071816 - cp's OEIS frontend

A077042 Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k(n-1)/2) and of x^ceiling(k(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139

Offset: 0

Henry Bottomley, Oct 22 2002

From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)

			Rows of square array start:
  1,    0,    0,    0,    0,    0,    0, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10,   20, ...
  1,    1,    3,    7,   19,   51,  141, ...
  1,    1,    4,   12,   44,  155,  580, ...
  1,    1,    5,   19,   85,  381, 1751, ...
  ...
Read by antidiagonals:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 1, 3, 3, 1, 1;
  0, 1, 6, 7, 4, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
Index entries for sequences related to compositions

Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779, A201550. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816, A133458.
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.

t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* _Jean-François Alcover, Nov 04 2011 *)

T(n,k)=if(n<1 || k<1,k==0,vecmax(Vec(((1-x^n)/(1-x))^k)))

By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).

T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012

A071817 Number of 3-digit numbers whose digits add up to n.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1

Offset: 1

Graeme McRae, Jun 07 2002

The sequence as a whole is palindromic; a(n) = a(28-n). - Jon E. Schoenfield, Nov 19 2016

			a(4) = 10 because there are 10 different 3-digit numbers whose digit sum is 4 (103, 112, 121, 130, 202, 211, 220, 301, 310, 400, which are the 3-digit elements of A052218).

Cf. A071816.

for i from 1 to 9*3 do a[i] := 0:od:for i from 100 to 999 dob := convert(i,base,10): s := sum(b[j],j=1..nops(b)):a[s] := a[s]+1:od:seq(a[j],j=1..3*9);

G.f.: (1 - x^10)^2*(x - x^10)/(1 - x)^3. - Miquel Cerda, Jul 09 2017

Corrected and extended by Sascha Kurz, Feb 07 2003

Name clarified by Jon E. Schoenfield, Nov 20 2016

A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.

Original entry on oeis.org

1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, 1, 6, 21, 56, 126, 246, 426, 666, 951, 1246, 1506, 1686, 1751, 1686, 1506, 1246, 951, 666, 426, 246, 126, 56, 21, 6, 1

Offset: 1

Juan Pablo Herrera P., Nov 18 2016

Sum of n-th row is n^6. The n-th row contains 6n-5 entries. Largest coefficients of each row are listed in A071816.
The n-th row is the sixth row of the n-nomial triangle.  For example, row 2 (1,6,15,20,15,6,1) is the sixth row in the binomial triangle
T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with six different playing cards. It is also the number of lattice paths from (0,0) to (6,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).

			Triangle starts:
1;
1, 6, 15, 20, 15, 6, 1;
1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1;
1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1.

Juan Pablo Herrera P., Rows n=1..50 of the triangle, flattened

Cf. A000332, A004737, A071816, A109439, A154286, A277949, A277950.

Mentioned in: A273975

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^6, {x, 0, 6 n}], x], {n, 10}] // Flatten

row(n) = Vec(((1 - x^n)/(1 - x))^6);
tabf(nn) = for (n=1, nn, print(row(n)));

T(n,k) = Sum_{i=k-n+1..k} A277950(T(n,i))

A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

Original entry on oeis.org

1, 35, 393, 2128, 8135, 24017, 60691, 134512, 273127, 512365, 908755, 1528688, 2473325, 3852919, 5832765, 8582336, 12354469, 17395119, 24072133, 32726960, 43874139, 57971221, 75715487, 97702640, 124853275, 157924585, 198105727

Offset: 1

Leonid Chindelevitch (leonidus(AT)mit.edu), Dec 22 2007

nonn

The middle coefficients for dimension d>=1 are in A000012, A000027, A077043, A005900, A077044, A071816, here, the d-th row in A077042.

For d=8 the sequence starts 1, 70, 1107, 8092, 38165, 135954, 398567, 1012664, 2306025, ... and for d=9 it starts 1, 126, 3139, 30276, 180325, 767394, 2636263, 7635987, 19610233, ... - R. J. Mathar, Sep 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. P. Stanley, Weyl groups, the Hard Lefschetz Theorem and the Sperner property, SIAM J. Alg. Disc. Meth. 1 (2) (1980) 168, see eq. (4).

Cf. A077043, A005900, A077044, A071816.

[-25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536 : n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

f:=(L,d)->(sum(x^k,k=0..L-1))^d; A:=[seq(coeff(f(j,7),x,floor(7*(j-1)/2)),j=1..25)];
A133458 := proc(n) -25/512*(-1)^n +2261/23040*n^2 -91/1536*(-1)^n*n^2 -77/1536*(-1)^n*n^4 +847/4608*n^4 +5887/11520*n^6 +25/512 ; end proc: # R. J. Mathar, Sep 05 2011

From R. J. Mathar, Feb 19 2010: (Start)
a(n)= 2*a(n-1) +4*a(n-2) -10*a(n-3) -5*a(n-4) +20*a(n-5) -20*a(n-7) +5*a(n-8) +10*a(n-9) -4*a(n-10) -2*a(n-11) +a(n-12).
G.f.: x*(1+33*x +319*x^2 +1212*x^3 +2662*x^4 +3320*x^5 +2662*x^6 +1212*x^7 +319*x^8 +33*x^9 +x^10)/ ((1+x)^5 * (1-x)^7).
a(n) = -25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536. (End)

More terms from R. J. Mathar, Feb 19 2010

A229735 151n^7/315+2n^5/9+7*n^3/45+n/7.

Original entry on oeis.org

0, 1, 70, 1107, 8092, 38165, 135954, 398567, 1012664, 2306025, 4816030, 9377467, 17232084, 30162301, 50651498, 82073295, 128912240, 197018321, 293897718, 429042211, 614299660, 864287973, 1196854978, 1633586615, 2200365864, 2927984825, 3852812366, 5017519755, 6471866692, 8273550157, 10489118490

Offset: 0

N. J. A. Sloane, Oct 01 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. B. Nathanson, Growth polynomials for additive quadruples and (h, k)-tuples, arXiv preprint arXiv:1305.7172, 2013. See Psi_4(n).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A071816.

[151*n^7/315+2*n^5/9+7*n^3/45+n/7: n in [0..30]]; // Vincenzo Librandi, Oct 06 2013

Table[151 n^7/315 + 2 n^5/9 + 7 n^3/45 + n/7, {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2013 *)

G.f.: x*(x^6+62*x^5+575*x^4+1140*x^3+575*x^2+62*x+1) / (x-1)^8. - Colin Barker, Oct 06 2013

A071009 Number of solutions (x,y,z,u,v,w) to x+y+z = u+v+w, 0<=x,y,z,u,v,w<=n-1, x>=y>=z, u>=v>=w.

Original entry on oeis.org

1, 4, 16, 48, 119, 256, 500, 900, 1525, 2456, 3796, 5664, 8207, 11588, 16004, 21672, 28845, 37800, 48856, 62356, 78691, 98280, 121592, 149128, 181445, 219132, 262840, 313256, 371131, 437256, 512492, 597740, 693977, 802224, 923580

Offset: 1

Vladeta Jovovic, Jun 10 2002

nonn

Cf. A071816.

Recurrence: a(n) = 3*a(n-1)-a(n-2)-4*a(n-3)+2*a(n-4)+2*a(n-5)+2*a(n-6)-4*a(n-7)-a(n-8)+3*a(n-9)-a(n-10). G.f.: x*(1+x+5*x^2+8*x^3+5*x^4+x^5+x^6)/(1+x+x^2)/(1+x)^2/(1-x)^6.

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A071817 Number of 3-digit numbers whose digits add up to n.

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A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.

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A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

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A229735 151*n^7/315+2*n^5/9+7*n^3/45+n/7.

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A071009 Number of solutions (x,y,z,u,v,w) to x+y+z = u+v+w, 0<=x,y,z,u,v,w<=n-1, x>=y>=z, u>=v>=w.

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A229735 151n^7/315+2n^5/9+7*n^3/45+n/7.