A287768 -id:A287768 - cp's OEIS frontend

A288842 Triangle (sans apex) of coefficients of terms of the form (eM_1)^j*(eM_2)^k re construction of triangle A287768.

1, 2, 3, 9, 6, 9, 36, 45, 18, 27, 135, 243, 189, 54, 81, 486, 1134, 1296, 729, 162, 243, 1701, 4860, 7290, 6075, 2673, 486, 729, 5832, 19683, 36450, 40095, 26244, 9477, 1458, 2187, 19683, 76545, 168399, 229635, 199017, 107163, 32805, 4374, 6561, 65610, 288684, 734832, 1194102, 1285956, 918540, 419904, 111537, 13122

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

			Triangle begins:
  1,  2;
  3,  9,  6;
  9, 36, 45, 18;

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=3, p.22, arXiv:1706.08381 [math.GM], 2017.

Columns are: A000244, A288834, A288835, A288836, A288838, etc.

Cf. A287768.

T[1, 1] = 1; T[1, 2] = 2;
T[n_, k_] /; 1 <= k <= n+1 := T[n, k] = 3 T[n-1, k-1] + 3 T[n-1, k];
T[, ] = 0;
Table[T[n, k], {n, 1, 9}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Nov 16 2018 *)

T(n+1,k+1) = 3*T(n,k) + 3*T(n,k+1).

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

Original entry on oeis.org

1, 1, 2, -1, 4, -3, 8, -8, 1, 16, -20, 5, 32, -48, 18, -1, 64, -112, 56, -7, 128, -256, 160, -32, 1, 256, -576, 432, -120, 9, 512, -1280, 1120, -400, 50, -1, 1024, -2816, 2816, -1232, 220, -11, 2048, -6144, 6912, -3584, 840, -72, 1, 4096, -13312, 16640, -9984

Offset: 0

N. J. A. Sloane

Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).
This triangle is generated from A118800 by shifting down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row. - Gary W. Adamson, Dec 16 2007
Unsigned triangle = A034839 * A007318. - Gary W. Adamson, Nov 28 2008
Triangle, with zeros omitted, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1, 1, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 16 2011
From Wolfdieter Lang, Aug 02 2014: (Start)
This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd-indexed columns removed.
If zeros are appended in each row n >= 1, in order to obtain a regular triangle (see the Philippe Deléham comment, g.f. and example) this becomes the Riordan triangle (1-x)/(1-2*x), -x^2/(1-2*x). See also the unsigned version A201701 of this regular triangle.
(End)
Apparently, unsigned diagonals of this array are rows of A200139. - Tom Copeland, Oct 11 2014
It appears that the coefficients are generated by the following: Let SM_k = Sum( d_(t_1, t_2)* eM_1^t_1 * eM_2^t_2) summed over all length 2 integer partitions of k, i.e., 1*t_1 + 2*t_2 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 2 data (i.e., SM_k = S_k/2 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(2,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2) form an irregular triangle, with one row for each k value starting with k=1. Thus this procedure and associated OEIS sequences A287768, A288199, A288207, A288211, A288245, A288188 are generalizations of Chebyshev polynomials of the first kind. - Gregory Gerard Wojnar, Jul 01 2017

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
The irregular triangle T(n,k) begins:
n\k     1      2     3      4     5     6   7   8 ....
0:      1
1:      1
2:      2     -1
3:      4     -3
4:      8     -8     1
5:     16    -20     5
6:     32    -48    18     -1
7:     64   -112    56     -7
8:    128   -256   160    -32     1
9:    256   -576   432   -120     9
10:   512  -1280  1120   -400    50    -1
11:  1024  -2816  2816  -1232   220   -11
12:  2048  -6144  6912  -3584   840   -72   1
13:  4096 -13312 16640  -9984  2912  -364  13
14:  8192 -28672 39424 -26880  9408 -1568  98  -1
15: 16384 -61440 92160 -70400 28800 -6048 560 -15
...
T(4,x) = 8*x^4 -8*x^2 + 1*x^0, T(5,x) = 16*x^5 - 20*x^3 + 5*x^1, with Chebyshev's T-polynomials (A053120). (End)
From _Philippe Deléham_, Dec 16 2011: (Start)
The triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) includes zeros and begins:
   1;
   1,   0;
   2,  -1,  0;
   4,  -3,  0,  0;
   8,  -8,  1,  0, 0;
  16, -20,  5,  0, 0, 0;
  32, -48, 18, -1, 0, 0, 0; (End)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From Rick L. Shepherd, Jul 06 2010]

Alois P. Heinz, Rows n = 0..200, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy] p. 795.
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=2, p. 22, arXiv:1706.08381 [math.GM], 2017.

Cf. A028298.
Reflection of A008310, the main entry. With zeros: A039991.
Cf. A053120 (row reversed table including zeros).
Cf. A118800, A034839, A081277, A124182.
Cf. A001333 (row sums 1), A001333 (alternating row sums). - Wolfdieter Lang, Aug 02 2014

b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

t[n_] := (Cos[n x] // TrigExpand) /. Sin[x]^m_ /; EvenQ[m] -> (1 - Cos[x]^2)^(m/2) // Expand; Flatten[Table[ r = Reverse @ CoefficientList[t[n], Cos[x]]; If[OddQ[Length[r]], AppendTo[r,0]]; Partition[r,2][[All, 1]],{n, 0, 13}] ][[1 ;; 53]] (* Jean-François Alcover, May 06 2011 *)
Tpoly[n_] := HypergeometricPFQ[{(1 - n)/2, -n/2}, {1/2}, 1 - x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

cos(n*x) = 2 * cos((n-1)*x) * cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). - Rick L. Shepherd, Jul 06 2010
G.f.: (1-x) / (1-2x+y*x^2). - Philippe Deléham, Dec 16 2011
Sum_{k=0..n} T(n,k)*x^k = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0, 1, 2, 3, 4, 5, 6, respectively. - Philippe Deléham, Dec 16 2011
T(n,k) = [x^k] hypergeom([1/2 - n/2, -n/2], [1/2], 1 - x). - Peter Luschny, Feb 03 2021
T(n,k) = (-1)^k * 2^(n-1-2*k) * A034807(n,k). - Hoang Xuan Thanh, Jun 21 2025

More terms from David W. Wilson

Row length sequence and link to Abramowitz-Stegun added by Wolfdieter Lang, Aug 02 2014

A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

Original entry on oeis.org

1, 8, -7, 64, -84, 21, 512, -896, 224, 196, -35, 4096, -8960, 2240, 3920, -350, -980, 35, 32768, -86016, 21504, 56448, -3360, -18816, 336, -5488, 1470, 1176, -21, 262144, -802816, 200704, 702464, -31360, -263424, 3136, -153664, 27440, 21952, -196, 38416, -1372, -3430, 7

Offset: 1

Gregory Gerard Wojnar, Jun 16 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_8)* eM_1^t_1 * eM_2^t_2 * ...*eM_8^t_8) summed over all length 8 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3+...+8*t_8=k, where SM_k are the averaged k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(8,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_8) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give: 1,8,85,932,10291,114878,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins
     1;
     8,    -7;
    64,   -84,   21;
   512,  -896,  224,  196,  -35;
  4096, -8960, 2240, 3920, -350, -980, 35;
  ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7). See Girard-Waring A210258. T(n,1)=8^(n-1)=A001018(n).

Java
```
// See Wojnar link.
```

A288199 Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.

Original entry on oeis.org

1, 4, -3, 16, -18, 3, 64, -96, 16, 18, -1, 256, -480, 80, 180, -30, -5, 1024, -2304, 384, 1296, -288, -108, -24, 9, 12, 4096, -10752, 1792, 8064, -2016, -1512, 112, 252, -112, 84, -7

Offset: 1

Gregory Gerard Wojnar, May 31 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, t_4)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3*eM_4^t_4) summed over all length 4 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + 4*t_4 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 4 data (i.e., SM_k = S_k/4 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(4,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, t_4) form an irregular triangle, with one row for each k value starting with k=1; "irregular" means that the number of terms in successive rows is nondecreasing.
Row sums of positive entries give 1, 4, 19, 98, 516, 2725, 14400.
Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins:
    1;
    4,   -3;
   16,  -18,   3;
   64,  -96,  16,  18, -1;
  256, -480,  80, 180, -5, -30;
  ...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 4*(eM_1)^2 - 3*eM_2;
Row 3: SM_3 = 16*(eM_1)^3 - 18*eM_1*eM_2 + 3*eM_3;
Row 4: SM_4 = 64*(eM_1)^4 - 96*(eM_1)^2*eM_2 + 16*eM_1*eM_3 + 18*(eM_2)^2 - 1*eM_4;
Row 5: SM_5 = 256*(eM_1)^5 - 480*(eM_1)^3*eM_2 + 80*(eM_1)^2*eM_2 + 180*eM_1*(eM_2)^2 - 30*eM_2*eM_3 - 5*eM_1*eM_4.

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=4, p.23, arXiv:1706.08381 [math.GM], 2017.

Cf. A210258, A028297 (m=2), A287768 (m=3), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).

A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

Original entry on oeis.org

1, 6, -5, 36, -45, 10, 216, -360, 80, 75, -10, 1296, -2700, 600, 1125, -250, -75, 5, 7776, -19440, 4320, 12150, -3600, -1125, -540, 225, 200, 36, -1, 46656, -136080, 30240, 113400, -37800, -23625, 2800, 5250, -3780, 3150, -350, 252, -105, -7

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(6,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_6) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,6,46,371,3026,24707,201748. Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins:
 1;
 6,-5;
 36,-45,10;
 216,-360,80,75,-10;
 1296,-2700,600,1125,-250,-75,5;
 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1;
 ...
Above represents:
 SM_1 = eM_1;
 SM_2 = 6*(eM_1)^2 - 5*eM_2;
 SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3;
 SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4;
 SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5;
 ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=6, p.24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring.

First column of triangle is powers of m=6, A000400.

Java
```
// See link.
```

A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

Original entry on oeis.org

1, 7, -6, 49, -63, 15, 343, -588, 140, 126, -20, 2401, -5145, 1225, 2205, -175, -525, 15, 16807, -43218, 10290, 27783, -1470, -8820, 126, -2646, 630, 525, -6, 117649, -352947, 84035, 302526, -12005, -108045, 1029, -64827, 10290, 8575, -49, 15435, -441, -1225, 1

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,7,64,609,5846,56161,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangular array begins...
1;
7,-6;
49,-63,15;
343,-588,140,126,-20;
2401,-5145,1225,2205,-175,-525,15;
16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6;
117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1;

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2107.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=5), A288211 (m=6), A288188 (m=8). Also see Girard-Waring A210258.

First entries of each row of triangle are powers of m=7, A000420.

Java
```
// See Wojnar link.
```

A288834 a(n) = (n+1) * 3^(n-1).

Original entry on oeis.org

2, 9, 36, 135, 486, 1701, 5832, 19683, 65610, 216513, 708588, 2302911, 7440174, 23914845, 76527504, 243931419, 774840978, 2453663097, 7748409780, 24407490807, 76709256822, 240588123669, 753145430616, 2353579470675, 7343167948506, 22876792454961

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

nonn

Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A000244, A027471, A287768.

Table[(n + 1)*3^(n - 1), {n, 27}] (* Michael De Vlieger, Jun 23 2017 *)
LinearRecurrence[{6,-9},{2,9},40] (* Harvey P. Dale, Dec 16 2018 *)

a(n) = (n+1)*3^(n-1) \\ Felix Fröhlich, Jun 19 2017

Vec((z*(2-3*z)/(1-3*z)^2) + O(z^30)) \\ Felix Fröhlich, Jun 19 2017

O.g.f.: z*(2-3*z)/(1-3*z)^2.
a(n) = -A287768(n+1,2).
a(n) = (n+1)*A000244(n-1). - Felix Fröhlich, Jun 19 2017
a(n) = A027471(n)/3 for n >= 3. - Art Baker, Apr 12 2019
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 9*log(3/2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 9*log(4/3). (End)

A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.

Original entry on oeis.org

1, 5, -4, 25, -30, 6, 125, -200, 40, 40, -4, 625, -1250, 250, 500, -25, -100, 1, 3125, -7500, 1500, 4500, -150, -1200, 6, -400, 60, 60, 15625, -43750, 8750, 35000, -875, -10500, 35, -7000, 700, 700, 1400, -14, -70

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, ... , t_5)* eM_1^t_1 * eM_2^t_2 *...* eM_5^t_5) summed over all length 5 integer partitions of k, i.e., 1*t_1+2*t_2+...+5*t_5=k, where SM_k are the averaged k-th power sum symmetric polynomials in 5 data (i.e., SM_k = S_k/5 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(5,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2,... , t_5) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

			Triangle begins:
    1;
    5,    -4;
   25,   -30,   6;
  125,  -200,  40,  40,   -4;
  625, -1250, 250, 500, -100, -25, 1;
  ...
Above represents:
SM_1 = 1*eM_1;
SM_2 = 5*(eM_1)^2 -4*eM_2;
SM_3 = 25*(eM_1)^3 - 30*eM_1*eM_2 + 6*eM_3;
SM_4 = 125*(eM_1)^4 - 200*(eM_1)^2*eM_2 + 40*eM_1*eM_3 + 40*(eM_2)^2 - 4*eM_4;
SM_5 = 625*(eM_1)^5 - 1250*(eM_1)^3*eM_2 + 250*(eM_1)^2*eM_3 + 500*eM_1*(eM_2)^2 - 100*eM_2*eM_3 - 25*eM_1*eM_4 + 1*eM_5;
...

Gregory Gerard Wojnar, Java program

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=6), A288245 (m=7), A288188 (m=8); A210258 Girard-Waring.

First column of triangle are powers of m=5, A000351.

Java
```
// See Java program link.
```

cp's OEIS Frontend

A288842 Triangle (sans apex) of coefficients of terms of the form (eM_1)^j*(eM_2)^k re construction of triangle A287768.

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A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

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A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

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A288199 Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.

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A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

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Java

A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

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Java

A288834 a(n) = (n+1) * 3^(n-1).

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Programs

Mathematica

PARI

PARI

Formula

A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.

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Examples

A288836 a(n) = (1/3!)3^(n+1)(n+5)(n+1)(n).

A288835 a(n) = (1/2!)3^n(n+3)*(n).