A238453 -id:A238453 - cp's OEIS frontend

A245798 Catalan number analogs for totienomial coefficients (A238453).

1, 1, 2, 4, 12, 36, 120, 360, 960, 3840, 13824, 41472, 152064, 506880, 2280960, 7983360, 29937600, 99792000, 266112000, 1197504000, 4790016000, 19160064000, 73156608000, 219469824000, 1009561190400, 3533464166400, 12563428147200, 54441521971200, 155547205632000

Offset: 0

Tom Edgar, Aug 22 2014

nonn

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Euler's totient function (A000010).
When the INTEGERS (2014) paper was written it was not known that this was an integral sequence (see the final paragraph of that paper). However, it is now known to be integral.
Another name could be phi-Catalan numbers. - Tom Edgar, Mar 29 2015

			We see that A238453(10,5) = 72 and A000010(5+1) = 2, so a(5) = (1/2)*72 = 36.

Tom Edgar, Table of n, a(n) for n = 0..28
Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A000010, A238453, A000108, A001088.

[(1/euler_phi(n+1))*prod(euler_phi(i) for i in [1..2*n])/prod(euler_phi(i) for i in [1..n])^2 for n in [0..100]]

a(n) = A238453(2*n,n) / A000010(n+1).

A272079 Row sums of A238453.

Original entry on oeis.org

1, 2, 3, 6, 10, 26, 30, 86, 154, 350, 514, 1542, 1930, 5882, 9086, 16242, 30098, 93474, 116174, 368318, 547218, 1050842, 1813030, 5703414, 7256850, 19610282, 33509306, 77004470, 122302250, 393465466, 399187282, 1302847262, 2380228834, 4947581482, 8604927394

Offset: 0

Tom Edgar, Apr 19 2016

nonn

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.

Cf. A000010, A238453.

P=[euler_phi(i) for i in [0..100]]
Tr=[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]]
print([sum(x) for x in Tr])

A255914 Triangle read by rows: T(n,k) = A007318(n,k)*A238453(n,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 24, 8, 1, 1, 20, 80, 80, 20, 1, 1, 12, 120, 160, 120, 12, 1, 1, 42, 252, 840, 840, 252, 42, 1, 1, 32, 672, 1344, 3360, 1344, 672, 32, 1, 1, 54, 864, 6048, 9072, 9072, 6048, 864, 54, 1, 1, 40, 1080, 5760, 30240, 18144, 30240

Offset: 0

Tom Edgar, Mar 10 2015

These are the generalized binomial coefficients associated with the sequence A002618.

			The first five terms in A002618 (n*phi(n)) are 1, 2, 6, 8, 20 and so T(4,2) = 8*6*2*1/((2*1)*(2*1)) = 24 and T(5,3) = 20*8*6*2*1/((6*2*1)*(2*1)) = 80.
The triangle begins:
1;
1, 1;
1, 2, 1;
1, 6, 6, 1;
1, 8, 24, 8, 1;
1, 20, 80, 80, 20, 1;
1, 12, 120, 160, 120, 12, 1;
1, 42, 252, 840, 840, 252, 42, 1

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A002618, A238453, A007318.

q=100 #change q for more rows
P=[i*euler_phi(i) for i in [0..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = Product_{i=1..n} A002618(i)/(Product_{i=1..k} A002618(i)*Product_{i=1..n-k} A002618(i)).

T(n,k) = A002618(n)/n*(k/A002618(k)*T(n-1,k-1)+(n-k)/A002618(n-k)*T(n-1,k)).

A272177 Alternating row sum of row 2n in A238453.

Original entry on oeis.org

1, 1, 2, 6, 42, 162, 634, 2726, 9330, 37814, 186034, 623934, 2370354, 10505738, 37417658, 113248562, 738713250, 2753640290, 7522970666, 46292418830, 146514514914, 496491671978, 2915741154442, 9464309679414, 39098814277474, 173211426442282, 652232674100690

Offset: 0

Tom Edgar, Apr 21 2016

nonn

The alternating row sums of odd rows in A238453 are 0.

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.

Cf. A000010, A238453, A272079.

P=[euler_phi(i) for i in [0..100]]
Tr=[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]]
L=[sum((-1)^i*x[i] for i in [0..len(x)-1]) for x in Tr]
print([y for y in L if y!=0])

a(n) = Sum_{k=0..2n}(-1)^k*A238453(2n,k).

A238688 Triangle read by rows: T(n,k) = A059381(n)/(A059381(k)*A059381(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 12, 32, 12, 1, 1, 24, 96, 96, 24, 1, 1, 24, 192, 288, 192, 24, 1, 1, 48, 384, 1152, 1152, 384, 48, 1, 1, 48, 768, 2304, 4608, 2304, 768, 48, 1, 1, 72, 1152, 6912, 13824, 13824, 6912, 1152, 72, 1, 1, 72, 1728, 10368, 41472, 41472

Offset: 0

Tom Edgar, Mar 02 2014

We assume that A059381(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_2 given in A007434.
Another name might be the 2-totienomial coefficients.

			The first five terms in the second Jordan totient function are 1,3,8,12,24 and so T(4,2) = 12*8*3*1/((3*1)*(3*1))=32 and T(5,3) = 24*12*8*3*1/((8*3*1)*(3*1))=96.
The triangle begins
1
1   1
1   3   1
1   8   8   1
1   12  32  12   1
1   24  96  96   24   1
1   24  192 288  192  24 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A007434, A059381, A238453.

q=100 #change q for more rows
P=[0]+[i^2*prod([1-1/p^2 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059381(n)/(A059381(k)* A059381(n-k)).
T(n,k) = prod_{i=1..n} A007434(i)/(prod_{i=1..k} A007434(i)*prod_{i=1..n-k} A007434(i)).
T(n,k) = A007434(n)/n*(k/A007434(k)*T(n-1,k-1)+(n-k)/A007434(n-k)*T(n-1,k)).

A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 6, 12, 12, 6, 1, 1, 12, 24, 36, 24, 12, 1, 1, 8, 32, 48, 48, 32, 8, 1, 1, 12, 32, 96, 96, 96, 32, 12, 1, 1, 12, 48, 96, 192, 192, 96, 48, 12, 1, 1, 18, 72, 216, 288, 576, 288, 216, 72, 18, 1, 1, 12, 72, 216, 432, 576, 576, 432, 216, 72, 12, 1

Offset: 0

Tom Edgar, Feb 27 2014

We assume that A175836(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Dedekind psi function A001615.
Another name might be the psi-nomial coefficients.

			The first five terms in the Dedekind psi function are 1,3,4,6,6 and so T(4,2) = 6*4*3*1/((3*1)*(3*1))=8 and T(5,3) = 6*6*4*3*1/((4*3*1)*(3*1))=12.
The triangle begins
1
1  1
1  3  1
1  4  4  1
1  6  8  6  1
1  6  12 12 6 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A001615, A175836, A238453.

a238498 n k = a238498_tabl !! n !! k
a238498_row n = a238498_tabl !! n
a238498_tabl = [1] : f [1] a001615_list where
   f xs (z:zs) = (map (div y) $ zipWith (*) ys $ reverse ys) : f ys zs
     where ys = y : xs; y = head xs * z
-- Reinhard Zumkeller, Mar 01 2014

A175836 := proc(n) option remember; local p;
`if`(n<2,1,n*mul(1+1/p,p=factorset(n))*A175836(n-1)) end:
A238498 := (n,k) -> A175836(n)/(A175836(k)*A175836(n-k)):
seq(seq(A238498(n,k),k=0..n),n=0..10); # Peter Luschny, Feb 28 2014

DedekindPsi[n_] := Sum[MoebiusMu[n/d] d^2 , {d, Divisors[n]}]/EulerPhi[n];
(* b = A175836 *) b[n_] := Times @@ DedekindPsi /@ Range[n];
T[n_, k_] := b[n]/(b[k] b[n-k]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jul 02 2019 *)

q=100 #change q for more rows
P=[0]+[i*prod([(1+1/x) for x in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A175836(n)/(A175836(k)*A175836(n-k)).
T(n,k) = prod_{i=1..n} A001615(i)/(prod_{i=1..k} A001615(i)*prod_{i=1..n-k} A001615(i)).
T(n,k) = A001615(n)/n*(k/A001615(k)*T(n-1,k-1)+(n-k)/A001615(n-k)*T(n-1,k)).
T(n,k) = A238688(n,k)/A238453(n,k).

A238743 Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 26, 26, 1, 1, 56, 208, 56, 1, 1, 124, 992, 992, 124, 1, 1, 182, 3224, 6944, 3224, 182, 1, 1, 342, 8892, 42408, 42408, 8892, 342, 1, 1, 448, 21888, 153216, 339264, 153216, 21888, 448, 1, 1, 702, 44928, 590976, 1920672, 1920672, 590976, 44928

Offset: 0

Tom Edgar, Mar 04 2014

We assume that A059382(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_3 given in A059376.
Another name might be the 3-totienomial coefficients.

			The first five terms in the third Jordan totient function are 1,7,26,56,124 and so T(4,2) = 56*26*7*1/((7*1)*(7*1))=208 and T(5,3) = 124*56*26*7*1/((26*7*1)*(7*1))=992.
The triangle begins
1
1 1
1 7   1
1 26  26   1
1 56  208  56   1
1 124 992  992  124  1
1 182 3224 6944 3224 182 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059382, A059376, A238688, A238453.

q=100 #change q for more rows
P=[0]+[i^3*prod([1-1/p^3 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059382(n)/(A059382(k)* A059382(n-k)).
T(n,k) = prod_{i=1..n} A059376(i)/(prod_{i=1..k} A059376(i)*prod_{i=1..n-k} A059376(i)).
T(n,k) = A059376(n)/n*(k/A059376(k)*T(n-1,k-1)+(n-k)/A059376(n-k)*T(n-1,k)).

A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 15, 1, 1, 80, 80, 1, 1, 240, 1280, 240, 1, 1, 624, 9984, 9984, 624, 1, 1, 1200, 49920, 149760, 49920, 1200, 1, 1, 2400, 192000, 1497600, 1497600, 192000, 2400, 1, 1, 3840, 614400, 9216000, 23961600, 9216000, 614400, 3840, 1, 1, 6480, 1658880

Offset: 0

Tom Edgar, Mar 04 2014

We assume that A059383(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_4 given in A059377.
Another name might be the 4-totienomial coefficients.

			The first five terms in the fourth Jordan totient function are 1,15,80,240,624 and so T(4,2) = 240*80*15*1/((15*1)*(15*1))=1280 and T(5,3) = 624*240*80*15*1/((80*15*1)*(15*1))=9984.
The triangle begins
1
1 1
1 15  1
1 80  80   1
1 240 1280 240  1
1 624 9984 9984 624 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059377, A059383, A238453, A238688, A238743.

q=100 #change q for more rows
P=[0]+[i^4*prod([1-1/p^4 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059383(n)/(A059383(k)* A059383(n-k)).
T(n,k) = prod_{i=1..n} A059377(i)/(prod_{i=1..k} A059377(i)*prod_{i=1..n-k} A059377(i)).
T(n,k) = A059377(n)/n*(k/A059377(k)*T(n-1,k-1)+(n-k)/A059377(n-k)*T(n-1,k)).

A239633 Triangle read by rows: T(n,k) = A059384(n)/(A059384(k)*A059384(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 31, 1, 1, 242, 242, 1, 1, 992, 7744, 992, 1, 1, 3124, 99968, 99968, 3124, 1, 1, 7502, 756008, 3099008, 756008, 7502, 1, 1, 16806, 4067052, 52501944, 52501944, 4067052, 16806, 1, 1, 31744, 17209344, 533489664, 1680062208, 533489664, 17209344, 31744

Offset: 0

Tom Edgar, Mar 22 2014

We assume that A059384(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_5 given in A059378.
Another name might be the 5-totienomial coefficients.

			The first five terms in the fifth Jordan totient function are 1,31,242,992,3124 and so T(4,2) = 992*242*31*1/((31*1)*(31*1))=7744 and T(5,3) = 3124*992*242*31*1/((242*31*1)*(31*1))=99968.
The triangle begins
1
1 1
1 31   1
1 242  242   1
1 992  7744  992   1
1 3124 99968 99968 3124 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059378, A059384, A238453, A238688, A238743, A238754.

q=100 #change q for more rows
P=[0]+[i^5*prod([1-1/p^5 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059384(n)/(A059384(k)* A059384(n-k)).
T(n,k) = prod_{i=1..n} A059378(i)/(prod_{i=1..k} A059378(i)*prod_{i=1..n-k} A059378(i)).
T(n,k) = A059378(n)/n*(k/A059378(k)*T(n-1,k-1)+(n-k)/A059378(n-k)*T(n-1,k)).

A239702 Triangle read by rows: T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 4, 4, 4, 1, 1, 2, 8, 4, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 1, 6, 6, 24, 6, 6, 1, 1, 1, 2, 2, 6, 12, 12, 6, 2, 2, 1, 1, 4, 8, 4, 24, 12, 24, 4, 8, 4, 1, 1, 10, 40, 40, 40, 60, 60, 40, 40, 40, 10, 1, 1, 2

Offset: 0

Tom Edgar, Mar 24 2014

A239682(0) = 1 since it is the empty product.

These are the generalized binomial coefficients associated with the sequence A173557.

			The first six terms A173557 are 1,1,2,1,4,2 and so T(4,2) = 1*2*1*1/((1*1)*(1*1))=2 and T(6,3) = 2*4*1*2*1*1/((2*1*1)*(2*1*1))=4.
The triangle begins
1
1 1
1 1  1
1 2  2  1
1 1  2  1  1
1 4  4  4  4  1
1 2  8  4  8  2 1
1 6 12 24 24 12 6 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A239682, A173557, A048804, A238453.

q=100 #change q for more rows
P=[0]+[prod([(x-1) for x in prime_divisors(n)]) for n in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] # generates the triangle up to q rows.

T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).
T(n,k) = prod_{i=1..n} A173557(i)/(prod_{i=1..k} A173557(i)*prod_{i=1..n-k} A173557(i)).
T(n,k) = A173557(n)/n*(k/A173557(k)*T(n-1,k-1)+(n-k)/A173557(n-k)*T(n-1,k)).

cp's OEIS Frontend

A245798 Catalan number analogs for totienomial coefficients (A238453).

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A272079 Row sums of A238453.

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A255914 Triangle read by rows: T(n,k) = A007318(n,k)*A238453(n,k).

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A272177 Alternating row sum of row 2n in A238453.

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A238688 Triangle read by rows: T(n,k) = A059381(n)/(A059381(k)*A059381(n-k)).

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A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)).

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A238743 Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

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A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

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