A051715 -id:A051715 - cp's OEIS frontend

A085737 Numerators in triangle formed from Bernoulli numbers.

1, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 2, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 8, -1, -1, 1, 0, 1, -1, 4, 4, -1, 1, 0, -1, 1, -1, -4, 8, -4, -1, 1, -1, 0, -1, 1, -8, 4, 4, -8, 1, -1, 0, 5, -5, 7, 4, -116, 32, -116, 4, 7, -5, 5, 0, 5, -5, 32, -28, 16, 16, -28, 32, -5, 5, 0

Offset: 0

N. J. A. Sloane, following a suggestion of J. H. Conway, Jul 23 2003

Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.

Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - Peter Luschny, May 04 2012

			Triangle of fractions begins
    1;
   1/2,   1/2;
   1/6,   1/3,   1/6;
    0,    1/6,   1/6,     0;
  -1/30,  1/30,  2/15,   1/30,  -1/30;
    0,   -1/30,  1/15,   1/15,  -1/30,    0;
   1/42, -1/42, -1/105,  8/105, -1/105, -1/42,   1/42;
    0,    1/42, -1/21,   4/105,  4/105, -1/21,   1/42,   0;
  -1/30,  1/30, -1/105, -4/105,  8/105, -4/105, -1/105, 1/30, -1/30;

Fabien Lange and Michel Grabisch, The interaction transform for functions on lattices Discrete Math. 309 (2009), no. 12, 4037-4048. [From _N. J. A. Sloane_, Nov 26 2011]
Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [_Peter Luschny_, May 04 2012]

Cf. A085738, A212196. See A051714/A051715 for another triangle that generates the Bernoulli numbers.

nmax:=11; for n from 0 to nmax do T(n, 0):= (-1)^n*bernoulli(n) od: for n from 1 to nmax do for k from 1 to n do  T(n, k) := T(n-1, k-1) - T(n, k-1) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(numer(T(n, k)), k=0..n), n=0..nmax);  # Johannes W. Meijer, Jun 29 2011, revised Nov 25 2012

t[n_, 0] := (-1)^n*BernoulliB[n]; t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1]; Table[t[n, k] // Numerator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014 *)

def BernoulliDifferenceTable(n) :
    def T(S, a) :
        R = [a]
        for s in S :
            a -= s
            R.append(a)
        return R
    def M(A, p) :
        R = T(A,0)
        S = add(r for r in R)
        return -S / (2*p+3)
    R = [1/1]
    A = [1/2,-1/2]; R.extend(A)
    for k in (0..n-2) :
        A = T(A,M(A,k)); R.extend(A)
        A = T(A,0); R.extend(A)
    return R
def A085737_list(n) : return [numerator(q) for q in BernoulliDifferenceTable(n)]
# Peter Luschny, May 04 2012

T(n, 0) = (-1)^n*Bernoulli(n), T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n.

T(n,k) = Sum_{j=0..k} binomial(k,j)*Bernoulli(n-j). [Lange and Grabisch]

Sign flipped in formula by Johannes W. Meijer, Jun 29 2011

A191302 Denominators in triangle that leads to the Bernoulli numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 15, 2, 6, 3, 2, 1, 5, 105, 2, 6, 15, 15, 2, 3, 3, 105, 105, 2, 2, 5, 7, 35, 2, 3, 3, 21, 21, 231, 2, 6, 15, 15, 21, 21, 2, 1, 5, 15, 1, 77, 15015, 2, 6, 3, 35, 15, 33, 1155

Offset: 0

Paul Curtz, May 30 2011

For the definition of the ASPEC array coefficients see the formulas; see also A029635 (Lucas triangle), A097207 and A191662 (k-dimensional square pyramidal numbers).
The antidiagonal row sums of the ASPEC array equal A042950(n) and A098011(n+3).
The coefficients of the T(n,m) array are defined in A190339. We define the coefficients of the SBD array with the aid of the T(n,n+1), see the formulas and the examples.
Multiplication of the coefficients in the rows of the ASPEC array with the coefficients in the columns of the SBD array leads to the coefficients of the BSPEC triangle, see the formulas. The BSPEC triangle can be looked upon as a spectrum for the Bernoulli numbers.
The row sums of the BSPEC triangle give the Bernoulli numbers A164555(n)/A027642(n).
For the numerators of the BSPEC triangle coefficients see A192456.

			The first few rows of the array ASPEC array:
  2, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  2, 5,  9, 14,  20,  27,   35,
  2, 7, 16, 30,  50,  77,  112,
  2, 9, 25, 55, 105, 182,  294,
The first few T(n,n+1) = T(n,n)/2 coefficients:
1/2, -1/6, 1/15, -4/105, 4/105, -16/231, 3056/15015, ...
The first few rows of the SBD array:
  1/2,   0,   0,     0
  1/2,   0,   0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15, -4/105
  1/2, -1/6, 1/15, -4/105
The first few rows of the BSPEC triangle:
  B(0) =   1   = 1/1
  B(1) =  1/2  = 1/2
  B(2) =  1/6  = 1/2 - 1/3
  B(3) =   0   = 1/2 - 1/2
  B(4) = -1/30 = 1/2 - 2/3 +  2/15
  B(5) =   0   = 1/2 - 5/6 +  1/3
  B(6) =  1/42 = 1/2 - 1/1 +  3/5  - 8/105
  B(7) =   0   = 1/2 - 7/6 + 14/15 - 4/15

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane) and A051714/A051715 (Akiyama-Tanigawa) for other triangles that lead to the Bernoulli numbers. - Johannes W. Meijer, Jul 02 2011

nmax:=13: mmax:=nmax:
A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end:
A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end:
for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od:
for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):=T(n-1,m+1)-T(n-1,m) od: od:
seq(T(n,n+1),n=0..nmax):
for n from 0 to nmax do ASPEC(n,0):=2: for m from 1 to mmax do ASPEC(n,m):= (2*n+m)*binomial(n+m-1,m-1)/m od: od:
for n from 0 to nmax do seq(ASPEC(n,m),m=0..mmax) od:
for n from 0 to nmax do for m from 0 to 2*mmax do SBD(n,m):=0 od: od:
for m from 0 to mmax do for n from 2*m to nmax do SBD(n,m):= T(m,m+1) od: od:
for n from 0 to nmax do seq(SBD(n,m), m= 0..mmax/2) od:
for n from 0 to nmax do BSPEC(n,2) := SBD(n,2)*ASPEC(2,n-4) od:
for m from 0 to mmax do for n from 0 to nmax do BSPEC(n,m) := SBD(n,m)*ASPEC(m,n-2*m) od: od:
for n from 0 to nmax do seq(BSPEC(n,m), m=0..mmax/2) od:
seq(add(BSPEC(n, k), k=0..floor(n/2)) ,n=0..nmax):
Tx:=0:
for n from 0 to nmax do for m from 0 to floor(n/2) do a(Tx):= denom(BSPEC(n,m)): Tx:=Tx+1: od: od:
seq(a(n),n=0..Tx-1); # Johannes W. Meijer, Jul 02 2011

(* a=ASPEC, b=BSPEC *) nmax = 13; a[n_, 0] = 2; a[n_, m_] := (2n+m)*Binomial[n+m-1, m-1]/m; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, nmax}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; dd = Diagonal[diff]; sbd[n_, m_] := If[n >= 2m, -dd[[m+1]], 0]; b[n_, m_] := sbd[n, m]*a[m, n-2m]; Table[b[n, m], {n, 0, nmax}, {m, 0, Floor[n/2]}] // Flatten // Denominator (* Jean-François Alcover_, Aug 09 2012 *)

ASPEC(n, 0) = 2 and ASPEC(n, m) = (2*n+m)*binomial(n+m-1, m-1)/m, n >= 0, m >= 1.
ASPEC(n, m) = ASPEC(n-1, m) + ASPEC(n, m-1), n >= 1, m >= 1, with ASPEC(n, 0) = 2, n >= 0, and ASPEC(0,m) = 1, m >= 1.
SBD(n, m) = T(m, m+1), n >= 2*m; see A190339 for the definition of the T(n, m).
BSPEC(n, m) = SBD(n, m)*ASPEC(m, n-2*m)
Sum_{k=0..floor(n/2)} BSPEC(n, k) = A164555(n)/A027642(n).

Edited, Maple program and crossrefs added by Johannes W. Meijer, Jul 02 2011

A051712 Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 25, 13, 9, 7, 29, 5, 31, 4, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 2, 49, 25, 17, 13, 53, 9, 55, 7, 19, 29, 59, 5, 61, 31, 21, 8, 65, 11, 67, 17, 23, 35, 71, 3, 73

Offset: 1

N. J. A. Sloane

			0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Cf. A026741, A045896, A051713.

Row 3 of table in A051714/A051715.

b[n_] := n/((n + 1) (n + 2)); Numerator[-Differences[Array[b, 100]]]
(* or *)
f[p_, e_] := p^e; f[2, e_] := If[e < 3, 1, 2^(e - 3)]; f[3, e_] := 3^(e - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n - 1]; Array[a, 100] (* Amiram Eldar, Nov 20 2022 *)

c(n) = a(n+1) is multiplicative with c(2^e) = 2^(e-3) if e > 2 and 1 otherwise, c(3^e) = 3^(e-1), and c(p^e) = p^e if p >= 5. [corrected by Amiram Eldar, Nov 20 2022]

Sum_{k=1..n} a(k) ~ (301/1152) * n^2. - Amiram Eldar, Nov 20 2022

A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 60, 60, 70, 84, 504, 120, 990, 165, 572, 1092, 2730, 280, 4080, 2448, 1938, 855, 7980, 1540, 10626, 3036, 4600, 7800, 17550, 819, 21924, 12180, 8990, 7440, 32736, 5984, 39270, 5355, 15540, 25308, 54834, 4940, 63960, 34440

Offset: 1

N. J. A. Sloane

			0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051712. Row 3 of table in A051714/A051715.

Denominator[#[[1]]-#[[2]]&/@(Partition[#[[1]]/(#[[2]]#[[3]])&/@Partition[ Range[50],3,1],2,1])] (* Harvey P. Dale, Nov 15 2014 *)

A362991 Triangle read by rows. T(n, k) = lcm{1, 2, ..., n+1} * Sum_{j=0..n-k} (-1)^(n-k-j) * j! * Stirling2(n - k, j) / (j + k + 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 0, 2, 3, 3, -2, 2, 9, 12, 12, 0, -2, 3, 8, 10, 10, 10, -10, -9, 24, 50, 60, 60, 0, 20, -30, -8, 50, 90, 105, 105, -84, 84, 18, -96, 0, 150, 245, 280, 280, 0, -84, 126, -24, -90, 18, 147, 224, 252, 252, 2100, -2100, 126, 1344, -600, -870, 343, 1568, 2268, 2520, 2520

Offset: 0

Peter Luschny, May 16 2023

A variant of the Akiyama-Tanigawa algorithm for the Bernoulli numbers A164555/ A027642.

			Triangle T(n, k) starts:
[0]   1;
[1]   1,   1;
[2]   1,   2,   2;
[3]   0,   2,   3,   3;
[4]  -2,   2,   9,  12,  12;
[5]   0,  -2,   3,   8,  10,  10;
[6]  10, -10,  -9,  24,  50,  60,  60;
[7]   0,  20, -30,  -8,  50,  90, 105, 105;
[8] -84,  84,  18, -96,   0, 150, 245, 280, 280;
[9]   0, -84, 126, -24, -90,  18, 147, 224, 252, 252;

Paolo Xausa, Table of n, a(n) for n = 0..11324 (rows 0..150 of the triangle, flattened)
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for sequences related to Bernoulli numbers.

Variant: A051714/A051715.

Cf. A362994 (column 0), A002944 (main diagonal), A164555/A027642 (Bernoulli).

LCM := n -> ilcm(seq((1 + i), i = 0..n)):
T := (n, k) -> LCM(n)*add((-1)^(n - k - j)*j!*Stirling2(n - k, j)/(j + k + 1), j = 0..n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

A362991row[n_]:=Table[LCM@@Range[n+1]Sum[(-1)^(n-k-j)j!StirlingS2[n-k,j]/(j+k+1),{j,0,n-k}],{k,0,n}];Array[A362991row,15,0] (* Paolo Xausa, Aug 09 2023 *)

def A362991Triangle(size):  # 'size' is the number of rows.
    A, T, l = [], [], 1
    for n in range(size):
        A.append(Rational(1/(n + 1)))
        for j in range(n, 0, -1):
            A[j - 1] = j * (A[j - 1] - A[j])
        l = lcm(l, n + 1)
        T.append([a * l for a in A])
    return T
A362991Triangle(10)

T(n, 0) = lcm(1, 2, ..., n+1) * Bernoulli(n, 1).

A244605 Numerators of the Akiyama-Tanigawa transform applied to 1/(n+1) with -1/2 instead of 1/2.

Original entry on oeis.org

1, 3, 19, 7, 449, 31, 2647, 127, 7649, 511, 67523, 2047, 11178659, 8191, 98305, 32767, 33419233, 131071, 209233981, 524287, 345855139, 2097151, 579668327, 8388607, 45565432859, 33554431, 411206281, 134217727, 209789384821, 536870911, 23993971665011, 2147483647, -5518887720767

Offset: 0

Paul Curtz, Jul 01 2014

The autosequence of the second kind A164555(n)/A027642(n) = 1, 1/2, 1/6, 0, -1/30, 0, ... (the second Bernoulli numbers) is the binomial transform of A027641(n)/A027642(n) = 1, -1/2, 1/6, 0, -1/30, 0, ... (the first Bernoulli numbers). Hence the name.
The Akiyama-Tanigawa transform applied to 1, -1/2, 1/3, 1/4, 1/5, 1/6, ... is:
     1,    -1/2,  1/3,  1/4,  1/5, ...
   3/2,    -5/3,  1/4,  1/5,  1/6, ...
  19/6,   -23/6, 3/20, 2/15, 5/42, ...
     7, -239/30, 1/20, 2/35, 5/84, ... .
The first column is a(n)/b(n) = 1, 3/2, 19/6, 7, 449/30, 31, 2647/42, 127, 7649/30, 511, 67523/66, 2047, ..., where the denominators are b(n) = A027642(n).
By the formula below, the Bernoulli numbers are linked to the Mersenne numbers A000225 (2^n-1).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A190339, A051715, A027641, A164555, A027642, A000225.

a[n_] := BernoulliB[n]+2^n-1 // Numerator; a[1] = 3; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jul 25 2014 *)

a(n) = my(b = numerator(bernfrac(n))/denominator(bernfrac(n))); if (n == 1, numerator(- b + 2^n - 1), numerator(b + 2^n - 1)); \\ Michel Marcus, Jul 18 2014

{a(n) = if( n<0, 0, 2*(n==1) + numerator( bernfrac(n) + 2^n - 1))}; /* Michael Somos, Aug 05 2014 */

a(n) = numerator of A164555(n)/A027642(n) + A000225(n).

a(12)-a(32) from Jean-François Alcover, Jul 01 2014

A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

Original entry on oeis.org

1, 1, -3, 1, -5, 5, 1, -7, 25, -5, 1, -9, 23, -35, 49, 1, -11, 73, -27, 112, -49, 1, -13, 53, -77, 629, -91, 58, 1, -15, 145, -130, 1399, -451, 753, -58, 1, -17, 95, -135, 2699, -2301, 8573, -869, 341, 1, -19, 241

Offset: 0

Paul Curtz, Apr 27 2012

In A190339 we saw that (the second Bernoulli numbers) A164555/A027642 is an eigensequence (its inverse binomial transform is the sequence signed) of the second kind, see A192456/A191302. We consider this array preceded by 1 for the second row, by 1, -3/2, for the third one; 1 is chosen and is followed by the differences of successive rows.
Hence
                    1    1/2   1/6      0
            1    -1/2   -1/3  -1/6  -1/30
      1   -3/2    1/6    1/6  2/15   1/15
  1 -5/2   5/3      0  -1/30 -1/15 -8/105.
The second row is A051716/A051717.
The (reduced) triangle before the square array (T(n,m) in A190339) is a(n)/b(n)=
B(0)=    1 = 1                 Redbernou1li
B(1)= -1/2 = 1  -3/2
B(2)=  1/6 = 1  -5/2  5/3
B(3)=    0 = 1  -7/2 25/6  -5/3
B(4)=-1/30 = 1  -9/2 23/3 -35/6  49/30
B(5)=    0 = 1 -11/2 73/6 -27/2 112/15 -49/30.
For the main diagonal, see A165142.
Denominator b(n) will be submitted.
This transform is valuable for every eigensequence of the second kind. For instance Leibniz's 1/n (A003506).
With increasing exponents for coefficients, polynomials CB(n,x) create Redbernou1li. See the formula.
Triangle Bernou1li for A027641/A027642 with the same denominator A080326 for every column is
1
1  -3/2
1  -5/2 10/6
1  -7/2 25/6 -10/6
1  -9/2 46/6 -35/6  49/30
1 -11/2 73/6 -81/6 224/30 -49/30.
For numerator by columns,see A000012, -A144396, A100536, Q(n)=n*(2*n^2+9*n+9)/2 , new.
Triangle Checkbernou1 with the same denominator A080326 for every row is
1/1
(2    -3)/2
(6   -15  +10)/6
(6   -21  +25  -10)/6
(30 -135 +230 -175  +49)/30
(30 -165 +365 -405 +224 -49)/30;
Hence for numerator: 1, 2-3, 16-15, 31-31, 309-310, 619-619, 8171-8166.
Absolute sum: 1, 5, 31, 62, 619, 1238, 17337. Reduced division by A080326:
1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, = A172030(n+1)/A172031(n+1).

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane), A051714/A051715 (Akiyama-Tanigawa), A192456/A191302 for other triangles that lead to the Bernoulli numbers.

CB(0,x) = 1,
CB(1,x) = 1 - 3*x/2,
CB(n,x) = (1-x)*CB(n-1,x) + B(n)*x^n , n > 1.

A230262 Numerators of Akiyama-Tanigawa algorithm applied to harmonic numbers, written by antidiagonals.

Original entry on oeis.org

1, 3, -1, 11, -2, 1, 25, -3, 1, 0, 137, -4, 3, 1, -1, 49, -5, 2, 1, -1, 0, 363, -6, 5, 2, -3, -1, 1, 761, -7, 3, 5, -1, -1, 1, 0, 7129, -8, 7, 5, 0, -4, 1, 1, -1, 7381, -9, 4, 7, 1, -1, -1, 1, -1, 0, 83711, -10, 9, 28, 49, -29, -5, 8, 1, -5, 5

Offset: 0

Paul Curtz, Nov 09 2013

Leading column gives the Bernoulli numbers: A027641(n)/A027642(n). In A051714, A164555 must be written instead of A027641.

			Numerators of
1,    3/2, 11/6, 25/12,...
-1/2, -2/3, -3/4,  -4/5,...
1/6,   1/6, 3/20,  2/15,... =A026741(n+1)/A045896(n+1)
0,    1/30, 1/20,  2/35,... =A194531/A193220.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, Vol 3 (2000), #00.2.9.

Cf. A051714/A051715, A001008/A002805.

t[1, k_] := HarmonicNumber[k]; t[n_, k_] := t[n, k] = k*(t[n-1, k] - t[n-1, k+1]); Table[t[n-k+1, k] // Numerator, {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 15 2013 *)

More terms from Jean-François Alcover, Nov 15 2013

cp's OEIS Frontend

A085737 Numerators in triangle formed from Bernoulli numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A191302 Denominators in triangle that leads to the Bernoulli numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A051712 Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A362991 Triangle read by rows. T(n, k) = lcm{1, 2, ..., n+1} * Sum_{j=0..n-k} (-1)^(n-k-j) * j! * Stirling2(n - k, j) / (j + k + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A244605 Numerators of the Akiyama-Tanigawa transform applied to 1/(n+1) with -1/2 instead of 1/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

Views

Author

Keywords

Comments

Crossrefs

Formula

A230262 Numerators of Akiyama-Tanigawa algorithm applied to harmonic numbers, written by antidiagonals.