A202369 -id:A202369 - cp's OEIS frontend

A178473 For n>=0, let n!^(4) = A202369(n+1) and, for 0<=m<=n, C^(4)(n,m) = n!^(4)/(m!^(4)*(n-m)!^(4)). The sequence gives triangle of numbers C^(4)(n,m) with rows of length n+1.

1, 1, 1, 1, 2, 1, 1, 273, 273, 1, 1, 68, 9282, 68, 1, 1, 55, 1870, 1870, 55, 1, 1, 546, 15015, 3740, 15015, 546, 1, 1, 29, 7917, 1595, 1595, 7917, 29, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

Conjecture. If p is prime of the form 4*k+1, then the k-th row contains two 1's and k-1 numbers multiple of p; if p is prime of the form 4*k+3, then the (2*k+1)-th row contains two 1's and 2*k numbers multiple of p.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1......1
.2..|..1......2......1
.3..|..1....273 ...273......1
.4..|..1.....68...9282.....68......1
.5..|..1.....55...1870...1870.....55......1
.6..|..1....546..15015...3740..15015....546....1
.7..|..1.....29...7917...1595...1595...7917...29.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917, A202941, A203484.

Conjecture. A007814(C^(4)(n,m)) = A007814(C(n,m)).

A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4Q^(4)(m-1)(i). For m>=0, the denominator for all 5m+1 terms of the m-th row is A202369(m+1).

Original entry on oeis.org

1, 6, 15, 10, 0, -1, 0, 36, 280, 795, 900, 88, -450, -20, 200, 1, -30, 0, 19656, 311220, 1991430, 6354075, 9367722, 1283100, -10854935, -1064700, 16237338, 615615, -16336320, -136500, 8189909, 8190, -1243800, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 23 2011

See comment in A175669.

			The sequence of polynomials begins
Q^(3)_0=1,
Q^(3)_1=(6*x^5+15*x^4+10*x^3-x)/30,
Q^(3)_2=(36*x^10+280*x^9+795*x^8+900*x^7+88*x^6-450*x^5-20*x^4+200*x^3+x^2-30*x)/1800.

Cf. A202339, A053657, A202367, A202368, A202369, A175699, A202717

Q^(4)_n(1)=1.

A175669 Triangle of numerators of coefficients of the polynomial Q^(2)m(n) defined by the recursion Q^(2)_0(n)=1; for m>=1, Q^(2)_m(n) = Sum{i=1..n} i^2Q^(2)_(m-1)(i). For m>=0, the denominator for all 3m+1 terms of the m-th row is A202367(m+1).

Original entry on oeis.org

1, 2, 3, 1, 0, 20, 96, 155, 90, 5, -6, 0, 280, 2772, 10518, 18711, 14385, 1323, -2863, -126, 360, 0, 2800, 47040, 323336, 1157760, 2238855, 2050020, 207158, -810600, -58505, 322740, 7956, -45360, 0, 12320, 314160, 3409472, 20401128, 72418826, 150057435, 154651321, 12413874, -101524412, -6408765, 82588957, 3394248, -37374084, -546480, 5443200, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 20 2011

Consider sequence of sequences of polynomials {Q^(0)_m(x)}, {Q^(1)_m(x)},...,{Q^(r)_m(x)},..., such that in every sequence m=0,1,...

Sequence {Q^(r)m(x)} is defined by the recursion: Q^(r)_0(x)=1; for m>=1 and integer x=n, Q^(r)_m(n)=sum{i=1,...,n}i^rQ^(r)(m-1)(i). By the induction, we see that polynomial Q^(r)_m(x) has degree (r+1)*m. Note that Q^(0)_m(n) is C(n+m-1,m), Q^(1)_m(n)=S(n+m,n), where S(k,l) are Stirling numbers of the second kind. Thus Q^(r)_m(x) is an r-generalization of binomial coefficients and Stirling numbers of the second kind. Moreover, for every r, LCM of denominators of the coefficients of Q^(r)_m(x) generate sequences of factorial type which possess important arithmetic properties. For r=0, it is n!, for r=1, it is A053657, for r=2,3,4 we obtain A202367, A202368, A202369. Denote the general term of the sequence corresponding to a given r by n!^(r) and, for 0<=m<=n, denote C^(r)(n,m)=n!^(r)/(m!^(r)*(n-m)!^(r). Then, for the "r-Pascal triangle", we have the following conjectural regularity: if a prime p==1 mod r, then the ((p-1)/r)-th row contains two 1's and numbers multiple of p. Cf. triangles A202917, A202941.

			The sequence of polynomials begins:
Q^(2)_0=1,
Q^(2)_1=(2*x^3+3*x^2+x)/6,
Q^(2)_2=(20*x^6+96*x^5+155*x^4+90*x^3+5*x^2-6*x)/360,
Q^(2)_3=(280*x^9+2772*x^8+10518*x^7+18711*x^6+14385*x^5+1323*x^4-2863*x^3 -126*x^2+360*x)/45360.

Cf. A202339, A053657, A202367, A202368, A202369.

Q^(2)_n(1)=1.

A202917 For n >= 0, let n!^(1) = A053657(n+1) and, for 0 <= m <= n, C^(1)(n,m) = n!^(1)/(m!^(1)*(n-m)!^(1)). The sequence gives a triangle of numbers C^(1)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 60, 10, 60, 1, 1, 1, 10, 10, 1, 1, 1, 126, 21, 1260, 21, 126, 1, 1, 1, 21, 21, 21, 21, 1, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 26 2011

1) Note that A053657(n+1) is the LCM of the denominators of the coefficients of the polynomials Q^(1)n(x) which, for integer x=k, are defined by the recursion Q^(1)_0(x)=1, for n>=1, Q^(1)_n(x) = Sum{i=1..k} i*Q^(1)(n-1)(i). Also note that Q^(1)_n(k) = S(k+n,k), where the numbers S(l,m) are Stirling numbers of the second kind. The sequence of polynomials {Q^(1)_n(x)} includes the family of sequences of polynomials {{Q^(r)_n}}(r>=0) described in a comment at A175669. In particular, the LCM of the denominators of the coefficients of Q^(0)_n(x) is n!.
2) This triangle differs from triangle A186430 which is defined according to the theory of factorials over sets by Bhargava. Unfortunately, this theory does not have a conversion theorem. Therefore it is not known if there is a set A such that n!^(1) = n!_A in the Bhargava sense.
3) If p is an odd prime, then the (p-1)-th row contains two 1's and p-2 numbers that are multiples of p. For a conjectural generalization, see comment in A175669.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....6.....1
.3..|..1.....1 ... 1  .....1
.4..|..1....60....10......60.....1
.5..|..1.....1....10......10.....1.....1
.6..|..1...126....21....1260....21...126.....1
.7..|..1.....1....21......21....21....21.....1.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; f1[n_] := A053657[n+1]; C1[n_, m_] := f1[n]/(f1[m] * f1[n-m]); Table[C1[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2016 *)

A007814(C^(1)(n,m)) = A007814(C(n,m)).

A202941 For n>=0, let n!^(2)=A202367(n+1) and, for 0<=m<=n, C^(2)(n,m)=n!^(2)/(m!^(2)*(n-m)!^(2)). The sequence gives triangle of numbers C^(2)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 21, 21, 1, 1, 20, 42, 20, 1, 1, 11, 22, 22, 11, 1, 1, 2730, 3003, 2860, 3003, 2730, 1, 1, 1, 273, 143, 143, 273, 1, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 26 2011

Conjecture. If p is an odd prime, then the ((p-1)/2)-th row contains two 1's and (p-3)/2 numbers multiple of p.

See also comments in A175669 and A202917.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1....10.....1
.3..|..1....21 ...21.....1
.4..|..1....20....42....20.....1
.5..|..1....11....22....22....11.....1
.6..|..1..2730..3003..2860..3003..2730.....1
.7..|..1.....1...273...143...143...273.....1.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917

If conjectural formula in A202367 is true, then A007814(C^(2)(n,m)) =A007814(C(n,m)).

A202368 LCM of denominators of the coefficients of polynomials Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1,Q^(3)_m(n)=sum{i=1,...,n}i^3*Q^(3)(m-1)(i).

Original entry on oeis.org

1, 4, 672, 13440, 58705920, 234823680, 11243357798400, 494707743129600, 4321766843980185600, 86435336879603712000, 450155234468976132096000

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2011

nonn

See comment in A175669.

Cf. A053657, A202367, A175669, A202369

A203484 For n>=0, let n!^(3) = A202368(n+1) and, for 0<=m<=n, C^(3)(n,m) = n!^(3)/(m!^(3)*(n-m)!^(3)). The sequence gives triangle of numbers C^(3)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 42, 1, 1, 5, 5, 1, 1, 1092, 130, 1092, 1, 1, 1, 26, 26, 1, 1, 1, 11970, 285, 62244, 285, 11970, 1, 1, 11, 3135, 627, 627, 3135, 11, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

Conjecture. If p is prime of the form 3*k+1, then the k-th row contains two 1's and k-1 numbers multiple of p; if p is prime of the form 3*k+2, then the (2*k+1)-th row contains two 1's and 2*k numbers multiple of p.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1......1
.2..|..1.....42.....1
.3..|..1......5 ....5......1
.4..|..1...1092...130...1092.....1
.5..|..1......1....26.....26.....1......1
.6..|..1..11970...285..62244...285..11970....1
.7..|..1.....11..3135....627...627...3135...11.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917, A202941.

Conjecture. A007814(C^(3)(n,m)) = A007814(C(n,m)).

A202717 Triangle of numerators of coefficients of the polynomial Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1, Q^(3)_m(n) = Sum{i=1...n} i^3*Q^(3)_(m-1)(i).

Original entry on oeis.org

1, 1, 2, 1, 0, 0, 21, 132, 294, 252, 21, -56, 0, 8, 0, 35, 450, 2293, 5700, 6405, 770, -3661, -240, 2320, 40, -672, 0, 0, 9555, 207480, 1889316, 9216312, 25051026, 33229560, 3678948, -35339304, -2666157, 51171120, 2178176, -49878192, -792064, 24460800, 4160, -3714816, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 23 2011

For m>=0, the denominator for all 4*m+1 terms of the m-th row is A202368(m+1).

See comment to A175669.

			The sequence of polynomials begins
Q^(3)_0=1,
Q^(3)_1=(x^4+2*x^3+x^2)/4,
Q^(3)_2=(21*x^8+132*x^7+294*x^6+252*x^5+21*x^4-56*x^3+8*x)/672,
Q^(3)_3=(35*x^12+450*x^11+2293*x^10+5700*x^9+6405*x^8+770*x^7-3661*x^6-240*x^5+2320*x^4+40x^3-672*x^2)/13440.

Cf. A202339, A053657, A202367, A202368, A202369, A175699.

Q^(3)_n(1)=1.

cp's OEIS Frontend

A178473 For n>=0, let n!^(4) = A202369(n+1) and, for 0<=m<=n, C^(4)(n,m) = n!^(4)/(m!^(4)*(n-m)!^(4)). The sequence gives triangle of numbers C^(4)(n,m) with rows of length n+1.

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A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4*Q^(4)(m-1)(i). For m>=0, the denominator for all 5*m+1 terms of the m-th row is A202369(m+1).

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A175669 Triangle of numerators of coefficients of the polynomial Q^(2)m(n) defined by the recursion Q^(2)_0(n)=1; for m>=1, Q^(2)_m(n) = Sum{i=1..n} i^2*Q^(2)_(m-1)(i). For m>=0, the denominator for all 3*m+1 terms of the m-th row is A202367(m+1).

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A202917 For n >= 0, let n!^(1) = A053657(n+1) and, for 0 <= m <= n, C^(1)(n,m) = n!^(1)/(m!^(1)*(n-m)!^(1)). The sequence gives a triangle of numbers C^(1)(n,m) with rows of length n+1.

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A202941 For n>=0, let n!^(2)=A202367(n+1) and, for 0<=m<=n, C^(2)(n,m)=n!^(2)/(m!^(2)*(n-m)!^(2)). The sequence gives triangle of numbers C^(2)(n,m) with rows of length n+1.

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A202368 LCM of denominators of the coefficients of polynomials Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1,Q^(3)_m(n)=sum{i=1,...,n}i^3*Q^(3)(m-1)(i).

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A203484 For n>=0, let n!^(3) = A202368(n+1) and, for 0<=m<=n, C^(3)(n,m) = n!^(3)/(m!^(3)*(n-m)!^(3)). The sequence gives triangle of numbers C^(3)(n,m) with rows of length n+1.

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A202717 Triangle of numerators of coefficients of the polynomial Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1, Q^(3)_m(n) = Sum{i=1...n} i^3*Q^(3)_(m-1)(i).

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A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4Q^(4)(m-1)(i). For m>=0, the denominator for all 5m+1 terms of the m-th row is A202369(m+1).

A175669 Triangle of numerators of coefficients of the polynomial Q^(2)m(n) defined by the recursion Q^(2)_0(n)=1; for m>=1, Q^(2)_m(n) = Sum{i=1..n} i^2Q^(2)_(m-1)(i). For m>=0, the denominator for all 3m+1 terms of the m-th row is A202367(m+1).