A331433 -id:A331433 - cp's OEIS frontend

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset: 0

N. J. A. Sloane, Jan 17 2020

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

			Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

[(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n,k), k=0..n),n=0..7); # Peter Luschny, Jan 18 2020
S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

T(n, 0) = (-1)^n*A000027(n+1).
T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
T(n, n-2) = A002738(n-2).
T(n, n-1) = (-1)*A002736(n).
T(n, n) = A002457(n).
T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
From Petros Hadjicostas, Jul 09 2020: (Start)
Michael Somos's formulas above can be restated as
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

Original entry on oeis.org

1, 3, 8, 6, 15, 27, 10, 24, 42, 64, 15, 35, 60, 90, 125, 21, 48, 81, 120, 165, 216, 28, 63, 105, 154, 210, 273, 343, 36, 80, 132, 192, 260, 336, 420, 512, 45, 99, 162, 234, 315, 405, 504, 612, 729, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, 66, 143, 231, 330

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), Oct 29 2004

Frank Buss gave this as a puzzle; K. L. Metlov solved it, submitting his result in the J language created by Kenneth Iverson. The program given below is only five tokens long. J defines a series of three functions to be a "fork" defined by x (f g h ) y = (x f y) g (f h y) - a generalization of the usual mathematical practice of writing (f + g) y to mean (f y) + (g y). J also has a primitive "half" and has a dummy function "cap" whose purpose is to permit more forks to be written. 3 (* * +) 5 is thus (3 * 5) * (3 + 5) or 120. cap half 3 (* * +) 5 is thus 60.
This sequence is the dimensions of the various irreducible representations of SU(3). In the language of physics, the integers m and n are one more than the numbers of quarks or antiquarks, respectively, that label the representation. - Alex Meiburg, Dec 13 2020 =
Comment on the previous one: D(n, m) = f(m+1, n+1) = (n+1)*(m+1)*(n+m+2), for 0 <= m <= n, (given as array D(n,m) as example in A212331) is the dimension of the irreducible SU(3) multiplet (n, m), denoted also by D(n, m). The multiplet (m, n) is denoted also by a bar over D(n, m). The irreducuble tensor t(n, m) is symmetric in n upper indices from {1,2,3}, symmetric in m lower indices, and traceless in every pair of an upper and a lower index. See the Coleman reference for a derivation. - Wolfdieter Lang, Dec 18 2020

			f(3, 5) is 60, from 1/2 * (3 * 5) * (3 + 5) or 1/2 * 15 * 8.
The triangle f(m, n) starts:
m\n     1   2   3   4   5   6   7   8   9   10   11 ...
1:      1
2:      3   8
3:      6  15  27
4:     10  24  42  64
5:     15  35  60  90 125
6:     21  48  81 120 165 216
7:     28  63 105 154 210 273 343
8:     36  80 132 192 260 336 420 512
9:     45  99 162 234 315 405 504 612 729
10:    55 120 195 280 375 480 595 720 855 1000
11:    66 143 231 330 440 561 693 836 990 1155 1331
... reformatted and extended by _Wolfdieter Lang_, Dec 18 2020

Sidney Coleman, Quantum Field Theory, Eds. Bryan Gin-ge Chen et al., World Scientific, 2019, eq. (37.8), p. 799.

Wikipedia, Clebsch-Gordan coefficients for SU(3)

Cf. A000217, A005563, A140091, A067728, A212331, A140681 (columns), A000578, A059270, A331433 (diagonals).
(diagonal).
See also A107985, A212331 (array as example).

J
```
cap half * * +
```

t[m_, n_] := (m*n)(m + n)/2; Flatten[ Table[ t[m, n], {m, 10}, {n, m}]] (* Robert G. Wilson v, Nov 04 2004 *)

f(m, n) = 1/2 * (m * n) * (m + n).

G.f.: x*y*(1 + 4*x*y + x^2*(y - 9)*y - 3*x^3*(y - 1)*y + 3*x^4*y^2)/((1 - x)^3*(1 - x*y)^4). - Stefano Spezia, Oct 01 2023

More terms from Robert G. Wilson v, Nov 04 2004

cp's OEIS Frontend

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

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A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

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cp's OEIS Frontend

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

J

Mathematica

Formula

Extensions

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.