A036970 -id:A036970 - cp's OEIS frontend

A065748 Triangle of Gandhi polynomial coefficients.

1, 1, 4, 6, 4, 15, 88, 220, 304, 250, 120, 28, 1025, 7308, 23234, 43420, 52880, 43880, 25088, 9680, 2340, 280, 209135, 1691024, 6237520, 13911400, 20956610, 22549360, 17853780, 10541440, 4639740, 1498280, 341000, 49920, 3640, 100482849

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 16 2001

First column is A064625.

			Triangle starts
1;
1,4,6,4;
15,88,220,304,250,120,28;
1025,...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. 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)

Cf. A064625, A036970

Let B(X, n) = X^4 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^4; then the (i, j)-th entry in the table is the coefficient of X^(5+j) in B(X, i).

A195505 Numerator of Sum_{k=1..n} H(k)/k^2, where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 11, 341, 2953, 388853, 403553, 142339079, 1163882707, 31983746689, 32452469713, 43725835522403, 44184852180503, 97954699428176291, 98731028315167091, 99421162547987123, 800313205356878959, 3953829021224881128767, 3973669953994085875967

Offset: 1

Franz Vrabec, Sep 19 2011

Lim_{n-> infinity} (a(n)/A195506(n)) = 2*Zeta(3) [L. Euler].

Sum_{k = 1..n} H(k)/k^2 is an example of a multiple harmonic (star) sum. Euler's result Sum_{k = 1..inf} H(k)/k^2 = 2*zeta(3) was the first evaluation of a multiple zeta star value. - Peter Bala, Jan 31 2019

			a(2) = 11 because 1 + (1 + 1/2)/2^2 = 11/8.
The first few fractions are 1, 11/8, 341/216, 2953/1728, 388853/216000, 403553/216000, 142339079/74088000, 1163882707/592704000, ... = A195505/A195506. - _Petros Hadjicostas_, May 06 2020

Seiichi Manyama, Table of n, a(n) for n = 1..768
Leonhard Euler, Meditationes circa singulare serierum genus, Novi. Comm. Acad. Sci. Petropolitanae, 20 (1775), 140-186.

Cf. A001008, A002117, A036970, A195506 (denominators).

s = 0; Table[s = s + HarmonicNumber[n]/n^2; Numerator[s], {n, 20}] (* T. D. Noe, Sep 20 2011 *)

H(n) = sum(k=1, n, 1/k);
a(n) = numerator(sum(k=1, n, H(k)/k^2)); \\ Michel Marcus, May 07 2020

From Peter Bala, Jan 31 2019: (Start)
Let S(n) = Sum_{k = 1..n} H(k)/k^2. Then
S(n) = 1 + (1 + 1/2^3)*(n-1)/(n+1) + (1/2^3 + 1/3^3)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^3 + 1/4^3)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ...
S(n)/n  = 1 + (1/2^4 - 1)*(n-1)/(n+1) + (1/3^4 - 1/2^4)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/4^4 - 1/3^4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ...
For odd n >= 3, 1/2*S((n-1)/2) = (n-1)/(n+1) + 1/2^3*(n-1)*(n-3)/((n+1)*(n+3)) + 1/3^3*(n-1)*(n-3)*(n-5)/((n+1)*(n+3)*(n+5)) + ....
Cf. A001008. See the Bala link in A036970. (End)

A058942 Triangle of coefficients of Gandhi polynomials.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 8, 22, 20, 6, 56, 184, 224, 120, 24, 608, 2248, 3272, 2352, 840, 120, 9440, 38080, 62768, 54336, 26208, 6720, 720, 198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040, 5410688, 24719488, 48207488, 52633344, 35371776

Offset: 1

David W. Wilson, Jan 12 2001

(1+x)^2 divides these polynomials for n > 2. - T. D. Noe, Jan 01 2008

			Triangle starts:
[1]
[1,      1]
[2,      4,      2]
[8,      22,     20,      6]
[56,     184,    224,     120,     24]
[608,    2248,   3272,    2352,    840,    120]
[9440,   38080,  62768,   54336,   26208,  6720,   720]
[198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040]

T. D. Noe, Rows n = 1..50 of triangle, flattened

First column is A005439, as are row sums. See also A036970.

Cf. A084938.

c[1][x_] = 1; c[n_][x_] :=  c[n][x] = (x+1)*((x+1)*c[n-1][x+1] - x*c[n-1][x]); Table[ CoefficientList[ c[n][x], x], {n, 9}] // Flatten (* Jean-François Alcover, Oct 09 2012 *)

# uses[delehamdelta from A084938]
def A058942_triangle(n) :
    A = [((i+1)//2)^2 for i in (1..n)]
    B = [((i+1)//2) for i in (1..n)]
    return delehamdelta(A, B)
A058942_triangle(10) # Peter Luschny, Nov 09 2019

C_1(x) = 1; C_n(x) = (x+1)*((x+1)*C_n-1(x+1) - x*C_n-1(x)).

Triangle T(n, k), read by rows; given by [1, 1, 4, 4, 9, 9, 16, 16, 25, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2005

A065747 Triangle of Gandhi polynomial coefficients.

Original entry on oeis.org

1, 1, 3, 3, 7, 30, 51, 42, 15, 145, 753, 1656, 1995, 1410, 567, 105, 6631, 39048, 100704, 149394, 140475, 86562, 34566, 8316, 945, 566641, 3656439, 10546413, 17972598, 20133921, 15581349, 8493555, 3246642, 841239, 135135, 10395

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 16 2001

First column is A064624.

			Triangle starts
1;
1, 3, 3;
7, 30, 51, 42, 15;
145, 753, 1656, 1995, 1410, 567, 105;
6631 ...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. 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)

Cf. A036970, A064624, A065748

Let B(X, n) = X^3 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^3; then the (i, j)-th entry is the table is the coefficient of X^(2+j) in B(X, i).

A065755 Triangle of Gandhi polynomial coefficients.

Original entry on oeis.org

1, 1, 5, 10, 10, 5, 31, 230, 755, 1440, 1760, 1430, 770, 260, 45, 6721, 60655, 250665, 628535, 1067865, 1299570, 1166945, 783720, 393855, 146025, 38500, 6630, 585, 5850271, 59885980, 285597890, 843288660, 1727996845, 2610132070, 3012643620

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001

First column is A065756. Second column is A065757.

			Irregular triangle begins:
1;
1,      5,  10,   10,    5;
31,   230, 755, 1440, 1760, 1430, 770, 260, 45;
6721, ...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.

Cf. A065756, A065757, A065747, A065748, A036970.

B[X_, 1] := X^5; B[X_, n_] := B[X, n] = X^5 (B[X+1, n-1] - B[X, n-1]) // Expand; row[1] = {1}; row[n_] := List @@ B[X, n] /. X -> 1; Array[row, 5] // Flatten (* Jean-François Alcover, Jul 08 2017 *)

Let B(X, n) = X^5 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^5; then the (i, j)-th entry in the table is the coefficient of X^(4+j) in B(X, i).

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A065748 Triangle of Gandhi polynomial coefficients.

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A195505 Numerator of Sum_{k=1..n} H(k)/k^2, where H(k) is the k-th harmonic number.

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A058942 Triangle of coefficients of Gandhi polynomials.

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A065747 Triangle of Gandhi polynomial coefficients.

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A065755 Triangle of Gandhi polynomial coefficients.

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