A272379 -id:A272379 - cp's OEIS frontend

A036970 Triangle of coefficients of Gandhi polynomials.

1, 1, 2, 3, 8, 6, 17, 54, 60, 24, 155, 556, 762, 480, 120, 2073, 8146, 12840, 10248, 4200, 720, 38227, 161424, 282078, 263040, 139440, 40320, 5040, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320, 28820619, 135634292

Offset: 1

N. J. A. Sloane

Another version of triangle T(n,k), 0 <= k <= n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] = 1; 0, 1; 0, 1, 2; 0, 3, 8, 6; 0, 17, 54, 60, 24; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 07 2004

			Triangle begins:
    1;
    1,   2;
    3,   8,   6;
   17,  54,  60,  24;
  155, 556, 762, 480, 120;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Peter Bala, The Gandhi polynomials as hypergeometric series
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
W. D. Cairns, Certain properties of binomial coefficients, Bull. Amer. Math. Soc. 26 (1920), 160-164. See p. 163 for a signed version.
Dominique Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.
Dominique Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
Marc Joye, Pascal Paillier and Berry Schoenmakers, On Second-Order Differential Power Analysis, in Cryptographic Hardware and Embedded Systems-CHES 2005, editors: Josyula R. Rao and Berk Sunar, Lecture Notes in Computer Science 3659 (2005) 293-308, Springer-Verlag.
Arthur Randrianarivony and Jiang Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.
Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, 40(2):175-180, May 2002.

First 2 columns are Genocchi numbers A001469, A005440, row sums are also A001469.

Cf. A083061, A272378, A272379, A272380.

B[1]:= X -> X^2:
for n from 2 to 12 do B[n]:= unapply(expand(X^2*(B[n-1](X+1)-B[n-1](X))),X) od:
seq(seq(coeff(B[i](X),X,1+j),j=1..i),i=1..12); # Robert Israel, Apr 21 2016

B[1][X_] = X^2;
B[n_][X_] := B[n][X] = X^2*(B[n-1][X+1] - B[n-1][X]) // Simplify;
Table[Coefficient[B[i][X], X, j+1], {i, 1, 12}, {j, 1, i}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

Let B(X, n) = X^2 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^2; then the (i, j)-th entry in the table is the coefficient of X^(1+j) in B(X, i). - Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001
From Gary W. Adamson, Jul 19 2011: (Start)
n-th row = top row of M^(n-1), M = an infinite square matrix in which the first "1" and right border of 1's of Pascal's triangle are deleted, as follows:
  1,  2,  0,  0,  0,  0, ...
  1,  3,  3,  0,  0,  0, ...
  1,  4,  6,  4,  0,  0, ...
  1,  5, 10, 10,  5,  0, ...
  1,  6, 15, 20, 15,  6, ...
  ...
(End)
Let G(n,x) = (-1)^(n+1)*B(-x,n). Then G(n,x) = (2*x/(x+1))*( 1 + 2^(2*n+1)*(x-1)/(x+2) + 3^(2*n+1)*(x-1)*(x-2)/((x+2)*(x+3)) + ... ). Cf. A083061. - Peter Bala, Feb 04 2019

More terms from David W. Wilson, Jan 12 2001

A272378 a(n) = n(6n^2 - 8*n + 3).

Original entry on oeis.org

0, 1, 22, 99, 268, 565, 1026, 1687, 2584, 3753, 5230, 7051, 9252, 11869, 14938, 18495, 22576, 27217, 32454, 38323, 44860, 52101, 60082, 68839, 78408, 88825, 100126, 112347, 125524, 139693, 154890, 171151, 188512, 207009, 226678, 247555, 269676, 293077

Offset: 0

Vincenzo Librandi, Apr 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A000384, A015237, A272379, A272380.

Magma
```
[n*(6*n^2 - 8*n + 3): n in [0..50]];
```

Table[n (6 n^2 - 8 n + 3), {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,1,22,99},40] (* Harvey P. Dale, Dec 29 2017 *)

vector(100, n, n--; n*(6*n^2 - 8*n + 3)) \\ Altug Alkan, Apr 29 2016

for n in range(0,10**3):print(n*(6*n**2-8*n+3),end=", ") # Soumil Mandal, Apr 30 2016

G.f.: x*(1 + 18*x + 17*x^2)/(1 - x)^4.
E.g.f.: x*(1 + 10*x + 6*x^2)*exp(x).
a(n) = n*A080859(n+1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3.
See page 7 in Brent's paper:
a(n) = n^2*A000384(n) - n*(n-1)*A000384(n-1).
A272379(n) = n^2*a(n) - n*(n-1)*a(n-1).
From Peter Bala, Jan 30 2019: (Start)
Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^7 + n^7)*a(n,x) and x*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^6 - n^6)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 7/2. See the Bala link in A036970. Cf. A272379. (End)

A272380 a(n) = n(120n^4 - 480n^3 + 762n^2 - 556*n + 155).

Original entry on oeis.org

0, 1, 342, 6315, 40492, 157125, 456546, 1099567, 2321880, 4448457, 7907950, 13247091, 21145092, 32428045, 48083322, 69273975, 97353136, 133878417, 180626310, 239606587, 313076700, 403556181, 513841042, 647018175, 806479752, 995937625, 1219437726, 1481374467

Offset: 0

Vincenzo Librandi, Apr 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A272378, A272379.

[n*(120*n^4 - 480*n^3 + 762*n^2 - 556*n + 155): n in [0..50]];

Table[n (120 n^4 - 480 n^3 + 762 n^2 - 556 n + 155), {n, 0, 50}]
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,342,6315,40492,157125},40] (* Harvey P. Dale, Mar 15 2018 *)

vector(100, n, n--; n*(120*n^4 - 480*n^3 + 762*n^2 - 556*n + 155)) \\ Altug Alkan, Apr 29 2016

O.g.f.: x*(1 + 336*x + 4278*x^2 + 7712*x^3 + 2073*x^4)/(1-x)^6.
E.g.f.: x*(1 + 170*x + 882*x^2 + 720*x^3 + 120*x^4)*exp(x).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), for n>5.
a(n) = n^2*A272379(n) - n*(n-1)*A272379(n-1), see page 7 in Brent's paper.
From Peter Bala, Jan 30 2019: (Start)
Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(120*x^4 - 480*x^3 + 762*x^2 - 556*x + 155) = Sum_{n >= 0} ((n+1)^11 + n^11)*a(n,x) and x*(120*x^4 - 480*x^3 + 762*x^2 - 556*x + 155)  = Sum_{n >= 0} ((n+1)^10 - n^10)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 11/2. See the Bala link in A036970. Cf. A272378 and A272379. (End)

cp's OEIS Frontend

A036970 Triangle of coefficients of Gandhi polynomials.

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A272378 a(n) = n(6n^2 - 8*n + 3).

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A272380 a(n) = n(120n^4 - 480n^3 + 762n^2 - 556*n + 155).

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cp's OEIS Frontend

A036970 Triangle of coefficients of Gandhi polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A272378 a(n) = n*(6*n^2 - 8*n + 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A272380 a(n) = n*(120*n^4 - 480*n^3 + 762*n^2 - 556*n + 155).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A272378 a(n) = n(6n^2 - 8*n + 3).

A272380 a(n) = n(120n^4 - 480n^3 + 762n^2 - 556*n + 155).