Franck Maminirina Ramaharo - cp's OEIS frontend

A386875 a(n) is the maximum number of strong sub-tournaments in an n-tournament.

0, 0, 0, 1, 3, 11, 27, 71, 159, 367, 783, 1695, 3519, 7359, 15039, 30847, 62463, 126719, 255231, 514559, 1033215, 2075647, 4160511, 8341503, 16703487, 33452031, 66949119, 133996543, 268091391, 536395775, 1073004543, 2146467839, 4293394431, 8587771903

Offset: 0

Franck Maminirina Ramaharo, Aug 06 2025

K. B. Reid and L. W. Beineke, "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 183 Corollary 6.2.

L. W. Beineke and F. Harary, The Maximum Number of Strongly Connected Subtournaments, Canadian Mathematical Bulletin, volume 8, Issue 4, 1965, pp. 491-498.
Index entries for linear recurrences with constant coefficients, signature (3,2,-12,4,12,-8).

Cf. A006918 (the maximum number of 3-cycles in an (n+2)-tournament).

Cf. A038376.

Table[If[Mod[n, 2] == 1, 2^n  - n*2^((n - 1)/2) - 1, 2^n - 3*n*2^((n - 4)/2) - 1], {n, 0, 20}]

a(n) := if mod(n, 2) = 1 then 2^n - n*2^((n - 1)/2) - 1 else 2^n - 3*n*2^((n - 4)/2) - 1$
makelist(a(n), n, 1, 20);

a(n) = 2^n - n*2^((n - 1)/2) - 1 if n is odd, and a(n) = 2^n - 3*n*2^((n - 4)/2) - 1 if n is even.
G.f.: x^3/((2*x-1)*(x-1)*(2*x^2-1)^2). - Alois P. Heinz, Aug 06 2025
E.g.f.: cosh(2*x) - cosh(x) - x*cosh(sqrt(2)*x) - sinh(x) + sinh(2*x) - 3*x*sinh(sqrt(2)*x)/(2*sqrt(2)). - Stefano Spezia, Aug 11 2025
a(2n+1) = A286778(n)/2. - R. J. Mathar, Aug 26 2025

A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).

Original entry on oeis.org

0, 1, 1, 0, 4, 7, 4, 1, 0, 15, 40, 42, 23, 7, 1, 0, 56, 201, 306, 262, 140, 48, 10, 1, 0, 209, 943, 1877, 2189, 1672, 881, 325, 82, 13, 1, 0, 780, 4239, 10412, 15368, 15276, 10841, 5660, 2194, 624, 125, 16, 1, 0, 2911, 18506, 54051, 96501, 118175, 105495

Offset: 1

Franck Maminirina Ramaharo, Aug 06 2025

T(n,k) is the number of ways to assign horizontal or vertical barriers at each interior construction dot of the 4 X 2n barrier-free Celtic shadow diagram CK_4^(2n) such that the resulting design consists of exactly k connected components.

The n-th row is the coefficients in the expansion of the Kauffman bracket polynomial for the shadow of the Celtic link CK_4^(2n).

			The triangle T(n,k) begins:
  n\k 0    1     2     3     4     5       6     7     8     9   10   11  12 13 14
  1:  0    1     1
  2:  0    4     7     4     1
  3:  0   15    40    42    23      7      1
  4:  0   56   201   306   262    140     48    10     1
  5:  0  209   943  1877  2189   1672    881   325    82    13    1
  6:  0  780  4239 10412 15368  15276  10841  5660  2194   624  125   16   1
  7:  0 2911 18506 54051 96501 118175 105495 71107 36885 14817 4579 1064 177 19  1
  ...

Roger Antonsen and Laura Taalman, Categorizing Celtic Knot Designs, in Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture, 2021, pp. 87-94.
Jonathan L. Gross and Thomas W. Tucker, A Celtic Framework for Knots and Links, Discrete & Computational Geometry 46 (2011), 86-99.
Franck Ramaharo, The bracket polynomial of the Celtic link shadow CK_4^(2n), arXiv:2508.10410 [math.GT], 2025. See p. 6.

Row sums: A013730.

Cf. A299989, A300192, A300453, A300454, A316659, A316989.

With[{nmax = 15}, CoefficientList[CoefficientList[Series[x*y*(x + 1)*(1 - x*y)/(1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2), {x, 0, 2*nmax}, {y, 0, nmax}], y], x]] // Flatten

nmax: 15$ v: x^2 + 4*x + 4$ w: sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12)$
p(n, x) := expand((1/(2*w))*(x^2 + x)*((((v + w)/2)^(n - 1))*(x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1))*(x^2 + 2*x + 4 - w)))$
create_list(ratcoef(p(n, x), x, k), n, 1, nmax, k, 0, 2*n);

T(n,1) = A001353(n).

G.f.: x*y*(x + 1)*(1 - x*y) / (1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2).

A323855 Triangle read by rows: T(n,k) is the denominator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).

Original entry on oeis.org

1, 2, 1, 6, 1, 1, 12, 12, 2, 1, 60, 4, 4, 1, 1, 20, 45, 8, 6, 2, 1, 140, 90, 120, 3, 3, 1, 1, 280, 5040, 80, 80, 2, 4, 2, 1, 2520, 1008, 378, 16, 144, 4, 12, 1, 1, 2520, 25200, 6048, 15120, 288, 240, 24, 3, 2, 1, 27720, 25200, 21600, 5040, 6048, 40, 240, 1, 2, 1, 1

Offset: 1

Franck Maminirina Ramaharo, Feb 01 2019

See A323854 for the definition of H(n,k).

			Triangle T(n,k) begins:
  n\k |   0    1    2    3    4    5    6
  ---------------------------------------
    1 |   1
    2 |   2    1
    3 |   6    1    1
    4 |  12   12    2    1
    5 |  60    4    4    3    1
    6 |  20   45    8    6    2    1
    7 | 140   90  120    3    3    1    1
    ...

Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic number identities and a related matrix representation, J. Korean Math. Soc, Volume 44, 2007, 487-498.
Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic numbers with Riordan arrays, Journal of Number Theory, Volume 128, Issue 2, 2008, 413-425.
Joseph M. Santmyer, A Stirling like sequence of rational numbers, Discrete Math., Volume 171, no. 1-3, 1997, 229-235, MR1454453.

Cf. A002805 (Column 0), A323854 (numerators).

H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
Table[Denominator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten

H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
create_list(denom(H(n, k)), n, 1, 20, k, 0, n - 1);

T(n, k) = denominator(substvec(diffop(L^(k+1)/X, [L, X], [1/X, 1], n), [L, X], [0, 1])/n!); \\ Jinyuan Wang, Mar 13 2025

T(n,k) = denominator of H(n,k), where H(n,k) = ((1/n!)*(-1)^(n+k+1))*(((d/dt)^n (1/t)*log(t)^(k+1))_{t=1}).

A323854 Triangle read by rows: T(n,k) is the numerator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).

Original entry on oeis.org

1, 3, 1, 11, 2, 1, 25, 35, 5, 1, 137, 15, 17, 3, 1, 49, 203, 49, 35, 7, 1, 363, 469, 967, 28, 23, 4, 1, 761, 29531, 801, 1069, 27, 39, 9, 1, 7129, 6515, 4523, 285, 3013, 75, 145, 5, 1, 7381, 177133, 84095, 341693, 8591, 7513, 605, 44, 11, 1, 83711, 190553, 341747, 139381, 242537, 1903, 10831, 33, 35, 6, 1

Offset: 1

Franck Maminirina Ramaharo, Feb 01 2019

Santmyer (1997) defined the generalized harmonic numbers H(n,k) of rank k by H(n,k) = Sum_{n_0 + n_1 + ... + n_k <= n} 1/(n_0*n_1*...*n_k).

If n >= 0, then the triangle {A323854(n+1,k)/A323855(n+1,k)}_{n,k} is the Riordan array (-log(1 - x)/(x*(1 - x)), -log(1 - x)/x).

			The triangle H(n,k) begins:
  n\k |   0        1       2       3     4      5     6
  -----------------------------------------------------
    1 |   1
    2 |   3/2      1
    3 |  11/6      2       1
    4 |  25/12    35/12    5/2     1
    5 | 137/60    15/4    17/4     3     1
    6 |  49/20   203/45   49/8    35/6   7/2   1
    7 | 363/140  469/90  967/120  28/3  23/3   4     1
    ...

Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic number identities and a related matrix representation, J. Korean Math. Soc, Volume 44, 2007, 487-498.
Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic numbers with Riordan arrays, Journal of Number Theory, Volume 128, Issue 2, 2008, 413-425.
Joseph M. Santmyer, A Stirling like sequence of rational numbers, Discrete Math., Volume 171, no. 1-3, 1997, 229-235, MR1454453.

Cf. A001008 (column 0), A323855 (denominators).

H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
Table[Numerator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten

H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
create_list(num(H(n, k)), n, 1, 20, k, 0, n - 1);

T(n, k) = -(-1)^(n+k)*numerator(substvec(diffop(L^(k+1)/X, [L, X], [1/X, 1], n), [L, X], [0, 1])/n!); \\ Jinyuan Wang, Mar 13 2025

T(n,k) = numerator of H(n,k), where H(n,k) = ((1/n!)*(-1)^(n+k+1))*(((d/dt)^n (1/t)*log(t)^(k+1))_{t=1}).

A306210 T(n,k) = binomial(n + k, n) - binomial(n + floor(k/2), n) + 1, square array read by descending antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 4, 1, 1, 3, 8, 7, 5, 1, 1, 4, 10, 17, 11, 6, 1, 1, 4, 16, 26, 31, 16, 7, 1, 1, 5, 19, 47, 56, 51, 22, 8, 1, 1, 5, 27, 65, 112, 106, 78, 29, 9, 1, 1, 6, 31, 101, 176, 232, 183, 113, 37, 10, 1, 1, 6, 41, 131, 296, 407, 435, 295, 157, 46, 11, 1

Offset: 0

Franck Maminirina Ramaharo, Jan 29 2019

There are at most T(n,k) possible values for the number of knots in an interpolatory cubature formula of degree k for an integral over an n-dimensional region.

			Square array begins:
  1, 1,  1,   1,   1,    1,    1,    1,     1,  ...
  1, 2,  2,   3,   3,    4,    4,    5,     5,  ...
  1, 3,  4,   8,  10,   16,   19,   27,    31,  ...
  1, 4,  7,  17,  26,   47,   65,  101,   131,  ...
  1, 5, 11,  31,  56,  112,  176,  296,   426,  ...
  1, 6, 16,  51, 106,  232,  407,  737,  1162,  ...
  1, 7, 22,  78, 183,  435,  841, 1633,  2794,  ...
  1, 8, 29, 113, 295,  757, 1597, 3313,  6106,  ...
  1, 9, 37, 157, 451, 1243, 2839, 6271, 12376,  ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  3,  1;
  1, 3,  4,  4,  1;
  1, 3,  8,  7,  5,  1;
  1, 4, 10, 17, 11,  6,  1;
  1, 4, 16, 26, 31, 16,  7, 1;
  1, 5, 19, 47, 56, 51, 22, 8, 1;
  ...

Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, A Survey of Methods for Constructing Cubature Formulae, In: Espelid T.O., Genz A. (eds), Numerical Integration, NATO ASI Series (Series C: Mathematical and Physical Sciences), Vol. 357, 1991, Springer, Dordrecht, pp. 1-24.
T. N. L. Patterson, On the Construction of a Practical Ermakov-Zolotukhin Multiple Integrator, In: Keast P., Fairweather G. (eds), Numerical Integration, NATO ASI Series (Series C: Mathematical and Physical Sciences), Vol. 203, 1987, Springer, Dordrecht, pp. 269-290.

Cf. A007318, A046854, A322596.

T[n_, k_] = Binomial[n + k, n] - Binomial[n + Floor[k/2], n] + 1;
Table[T[k, n - k], {k, 0, n}, {n, 0, 20}] // Flatten

T(n, k) := binomial(n + k, n) - binomial(n + floor(k/2), n) + 1$
create_list(T(k, n - k), n, 0, 20, k, 0, n);

T(n,k) = A007318(n+k,n) - A046854(n+k,n) + 1.

G.f.: (1 - x - x^2 + x^3 - 2*y + 2*x*y + y^2 - x*y^2 + x^2*y^2)/((1 - x)*(1 - y)*(1 - x - y)*(1 - x^2 - y)).

A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 9, 7, 4, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 5, 12, 21, 26, 21, 12, 5, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 6, 19, 42, 66, 77, 66, 42, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Jan 22 2019

For n >= 1, T(n,k) is the number of nodes in n-dimensional space for Mysovskikh's cubature formula which is exact for any polynomial of degree k of n variables.

			Array begins:
  1, 1, 1,  1,  1,   1,   1,    1,    1,    1, ...
  1, 1, 2,  2,  3,   3,   4,    4,    5,    5, ...
  1, 1, 2,  4,  5,   7,  10,   12,   15,   19, ...
  1, 1, 3,  5,  9,  14,  21,   30,   42,   55, ...
  1, 1, 3,  7, 14,  26,  42,   66,   99,  143, ...
  1, 1, 4, 10, 21,  42,  77,  132,  215,  334, ...
  1, 1, 4, 12, 30,  66, 132,  246,  429,  715, ...
  1, 1, 5, 15, 42,  99, 215,  429,  805, 1430, ...
  1, 1, 5, 19, 55, 143, 334,  715, 1430, 2702, ...
  1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  5,  3,  1,  1;
  1, 4,  7,  9,  7,  4,  1,  1;
  1, 4, 10, 14, 14, 10,  4,  1, 1;
  1, 5, 12, 21, 26, 21, 12,  5, 1, 1;
  1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
  ...

Ronald Cools, Encyclopaedia of Cubature Formulas
Ivan P. Mysovskikh, On the construction of cubature formulae for very simple domains, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.

Main diagonal: A000108.

Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.

b(n, k) := (n + k)!/((n + 1)!*k!)$
T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
create_list(T(k, n - k), n, 0, 15, k, 0, n);

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.

Original entry on oeis.org

1, 7, 17, 39, 81, 151, 257, 407, 609, 871, 1201, 1607, 2097, 2679, 3361, 4151, 5057, 6087, 7249, 8551, 10001, 11607, 13377, 15319, 17441, 19751, 22257, 24967, 27889, 31031, 34401, 38007, 41857, 45959, 50321, 54951, 59857, 65047, 70529, 76311, 82401, 88807

Offset: 0

Franck Maminirina Ramaharo, Jan 23 2019

For n >= 2, a(n) gives the number of function evaluations for Dooren and Ridder's degree 5 and 7 cubature rule over an n-dimensional cube, with the exception of a(3) = 45 and a(4) = 97.

Ronald Cools, Encyclopaedia of Cubature Formulas
Paul van Dooren and Luc de Ridder, An adaptive algorithm for numerical integration over an n-dimensional cube, Journal of Computational and Applied Mathematics, Vol. 2 (1976), 207-217.
Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics, Vol. 6 (1980), 295-302.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: 2*A093328.

Cf. A174794, A321124, A322594, A322595.

[(4*n^3-6*n^2+20*n+3)/3$n=0..50]; # Muniru A Asiru, Jan 23 2019

Table[(4*n^3 - 6*n^2 + 20*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 - 6*n^2 + 20*n + 3)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
G.f.: (1 + 3*x - 5*x^2 + 9*x^3)/((1 - x)^4).
E.g.f.: (1/3)*(3 + 18*x + 6*x^2 + 4*x^3)*exp(x).

A321122 a(n) = n-th row common denominator of A321121.

Original entry on oeis.org

4, 2, 3, 8, 36, 96, 44, 360, 492, 448, 1836, 5016, 2284, 18720, 25572, 23288, 95436, 260736, 118724, 973080, 1329252, 1210528, 4960836, 13553256, 6171364, 50581440, 69095532, 62924168, 257868036, 704508576, 320792204, 2629261800, 3591638412, 3270846208

Offset: 0

Franck Maminirina Ramaharo, Nov 21 2018

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.3.

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II. The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.

Cf. A321121 (Numerators).

Cf. A002176, A093736, A100621, A100646, A100648, A321119.

s = -2 + Sqrt[3];
e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));
f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));
w[n_, k_] := If[k == 0 || k == n, 1/4 + e[n]/6, If[k == 1 || k == n - 1, 2 - (1 + 1/6)*e[n], 1 + f[n, k]/4]];
a[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}];
Join[{4, 3, 2}, Table[a[n], {n, 3, 50}]]

s : -2 + sqrt(3)$
e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$
f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$
w(n, k) := if k = 0 or k = n then 1/4 + e(n)/6 else if k = 1 or k = n - 1 then 2 - (1 + 1/6)*e(n) else 1 + f(n, k)/4$
a(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
append([4, 2, 3], makelist(a(n), n, 3, 50));

Let s = -2 + sqrt(3), and define e(n) = s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n)), f(n,k) = 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n)), and w(n,0) = 1/4 + e(n)/6, w(n,1) = 2 - (1 + 1/6)*e(n), w(n,k) = 1 + f(n,k)/4 for 2 <= k <= n - 2. Then a(n) = LCM of denominators of {w(n,k), 0 <= k <= n} for n >= 3.
a(n) = 52*a(n-6) - a(n-12) for n >= 15 (conjectured).
G.f.: (4 + 2*x + 3*x^2 + 8*x^3 + 36*x^4 + 96*x^5 - 164*x^6 + 256*x^7 + 336*x^8 + 32*x^9 - 36*x^10 + 24*x^11 + 2*x^13 - 9*x^14)/(1 - 52*x^6 + x^12) (conjectured).

A321121 Triangle read by rows: T(n,k) is the unreduced numerator of the k-th weight in the quadrature rule for parabolic runout spline with respect to a mesh of n + 1 points.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 3, 9, 9, 3, 13, 44, 30, 44, 13, 35, 115, 90, 90, 115, 35, 16, 53, 40, 46, 40, 53, 16, 131, 433, 330, 366, 366, 330, 433, 131, 179, 592, 450, 504, 486, 504, 450, 592, 179, 163, 539, 410, 458, 446, 446, 458, 410, 539, 163, 668, 2209, 1680, 1878, 1824, 1842, 1824, 1878, 1680, 2209, 668

Offset: 0

Franck Maminirina Ramaharo, Nov 16 2018

The weights in this quadrature rule are T(n,k)/A321122(n), 0 <= k <= n. For n = 1, 2, 3, we obtain the trapezoid rule, Simpson's rule, and Simpson's 3/8 rule, respectively.

			Triangle begins (denominator is factored out):
    0;                                                 1/4
    1,   1;                                            1/2
    1,   4,   1;                                       1/3
    3,   9,   9,   3;                                  1/8
   13,  44,  30,  44,  13;                             1/36
   35, 115,  90,  90, 115,  35;                        1/96
   16,  53,  40,  46,  40,  53,  16;                   1/44
  131, 433, 330, 366, 366, 330, 433, 131;              1/360
  179, 592, 450, 504, 486, 504, 450, 592, 179;         1/492
  163, 539, 410, 458, 446, 446, 458, 410, 539, 163;    1/448
  ...

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.3.

Franck Maminirina Ramaharo, Rows n = 0..150 of triangle, flattened
Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II. The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.
Wikipedia, Newton-Cotes formulas

Cf. A321122 (Common denominators).

Cf. A093735/A093736 (Newton-Cotes formulas), A100640/A100641 (Cotesian numbers), A321118/A321119 (Holladay-Sard best quadrature formulas).

s = -2 + Sqrt[3];
e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));
f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));
w[n_, k_] := If[k == 0 || k == n, 1/4 + e[n]/6, If[k == 1 || k == n - 1, 2 - (1 + 1/6)*e[n], 1 + f[n, k]/4]];
a321122[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}]
Join[{0, 1, 1, 1, 4, 1}, Table[FullSimplify[a321122[n]*w[n, k]], {n, 3, 12}, {k, 0, n}]] // Flatten

s : -2 + sqrt(3)$
e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$
f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$
w(n, k) := if k = 0 or k = n then 1/4 + e(n)/6 else if k = 1 or k = n - 1 then 2 - (1 + 1/6)*e(n) else 1 + f(n, k)/4$
a321122(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
append([0, 1, 1, 1, 4, 1], create_list(fullratsimp(a321122(n)*w(n, k)), n, 3, 12, k, 0, n));

T(n,k) = T(n,n-k).
T(0,0) = 0 and T(n,k) = A093735(n,k) for n = 1, 2, 3.
Let s = -2 + sqrt(3), and define e(n) = s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n)), f(n,k) = 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n)), and w(n,0) = 1/4 + e(n)/6, w(n,1) = 2 - (1 + 1/6)*e(n), w(n,k) = 1 + f(n,k)/4 for 2 <= k <= n - 2. Then T(n,k) = A321122(n)*w(n,k) for 0 <= k <= n, n >= 3.

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

Original entry on oeis.org

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

Offset: 0

Franck Maminirina Ramaharo, Dec 19 2018

For n >= 6, a(n) is the number of evaluating points on the hypersphere in R^n in Stoyanovas's degree 7 cubature rule.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Srebra B. Stoyanova, Cubature of the seventh degree of accuracy for the hypersphere, Journal of Computational and Applied Mathematics Vol. 84 (1997), 15-21.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: A027693.

Cf. A000292, A161680, A174794, A321124, A322594.

[(n^3 + 9*n + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019

Table[(n^3 + 9*n + 14*n + 9)/3, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{3,11,21,35},50] (* Harvey P. Dale, Aug 19 2020 *)

makelist((n^3 + 9*n + 14*n + 9)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = 2*binomial(n + 1, 3) + 6*binomial(n + 1, 2) + 2*binomial(n + 1, 1) + 1.
G.f.: (3 - x - 5*x^2 + 5*x^3)/(1 - x)^4. [Corrected by Georg Fischer, May 23 2019]
E.g.f.: (1/3)*(9 + 24*x + 12*x^2 + x^3)*exp(x).

cp's OEIS Frontend

User: Franck Maminirina Ramaharo

A386875 a(n) is the maximum number of strong sub-tournaments in an n-tournament.

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A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2*w)) * (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1)) * (x^2 + 2*x + 4 - w)), where v = x^2 + 4*x + 4 and w = sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12).

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A323855 Triangle read by rows: T(n,k) is the denominator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).

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A323854 Triangle read by rows: T(n,k) is the numerator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).

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A306210 T(n,k) = binomial(n + k, n) - binomial(n + floor(k/2), n) + 1, square array read by descending antidiagonals (n >= 0, k >= 0).

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A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

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A322597 a(n) = (4*n^3 - 6*n^2 + 20*n + 3)/3.

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A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.