A116672 -id:A116672 - cp's OEIS frontend

A116673 Row sums of triangle A116672, in which the binomial transform of the n-th row lists the Euler transform of the n-th sequence in A007318 (Pascal's Triangle).

1, 2, 4, 10, 26, 80, 262, 950

Offset: 1

Alford Arnold, Feb 22 2006

nonn

A116673 is to A096807 as Table A116672 is to Table A096806. The difference between the two tables is of historical interest. (cf. A096751 and A007326).

			A116672 begins
1; 1,1; 1,2,1; 1,4,4,1; 1,6,11,7,1; 1,10,27,29,12,1; 1,14,57,96,72,21,1; 1,21,117,277,319,176,38,1; . . . so
A116673 begins 1 2 4 10 26 80 262 950 ...

Cf. A007318, A007326, A096751, A096806, A096807, A116672.

A289656 Triangle read by rows: row n gives the first n terms of the binomial transform of the n-th row of A116672.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 5, 13, 26, 1, 7, 24, 59, 120, 1, 11, 48, 141, 331, 672, 1, 15, 86, 310, 855, 1982, 4067, 1, 22, 160, 692, 2214, 5817, 13301, 27428

Offset: 1

N. J. A. Sloane, Jul 19 2017

Rows 4, 5, 6 match the starts of sequences A008778, A008779, A008780.

			Triangle begins:
[1]
[1, 2]
[1, 3, 6]
[1, 5, 13, 26]
[1, 7, 24, 59, 120]
[1, 11, 48, 141, 331, 672]
[1, 15, 86, 310, 855, 1982, 4067]
[1, 22, 160, 692, 2214, 5817, 13301, 27428]
...

N. J. A. Sloane, Transforms

Cf. A116672, A008778, A008779, A008780.

A008778 a(n) = (n+1)(n^2 +8n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 5, 13, 26, 45, 71, 105, 148, 201, 265, 341, 430, 533, 651, 785, 936, 1105, 1293, 1501, 1730, 1981, 2255, 2553, 2876, 3225, 3601, 4005, 4438, 4901, 5395, 5921, 6480, 7073, 7701, 8365, 9066, 9805, 10583, 11401, 12260, 13161, 14105, 15093, 16126, 17205, 18331

Offset: 0

N. J. A. Sloane

Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i-1,j)-m(i-1,j-1); then a(n)=m(n+3,3) - Benoit Cloitre, May 08 2002
a(n) = number of (n+6)-bit binary sequences with exactly 6 1's none of which is isolated. - David Callan, Jul 15 2004
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Sum of first n triangular numbers plus previous triangular number. - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
a(n) = Sum of first (n+1) triangular numbers plus n-th triangular number (see penultimate formula by Henry Bottomley). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
For n > 0, a(n-1) is the number of compositions of n+6 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
The binomial transform of [1,4,4,1,0,0,0,...], the 4th row in A116672. - R. J. Mathar, Jul 18 2017

			G.f. = 1 + 5*x + 13*x^2 + 26*x^3 + 45*x^4 + 71*x^5 + 105*x^6 + 148*x^7 + 201*x^8 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190 eq. (11.4.7).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 15.
Milan Janjic, Two Enumerative Functions
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A022811-A022817, A024207-A024210.
Column 1 of triangle A094415.
Row n=4 of A022818.
Cf. A002411, A008779, A005712 (partial sums), A034856 (first diffs).

List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); # G. C. Greubel, Sep 11 2019

[(n+1)*(n^2+8*n+6)/6: n in [0..50]]; // Vincenzo Librandi, May 21 2011

seq(1+4*k+4*binomial(k, 2)+binomial(k, 3), k=0..45);

Table[(n+1)*(n^2+8*n+6)/6, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009, modified by G. C. Greubel, Sep 11 2019 *)
LinearRecurrence[{4,-6,4,-1}, {1,5,13,26}, 51] (* G. C. Greubel, Sep 11 2019 *)

Vec((1+x-x^2)/(1-x)^4 + O(x^50)) \\ Altug Alkan, Jan 07 2016

[(n+1)*(n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Sep 11 2019

a(n) = dot_product(n, n-1, ...2, 1)*(2, 3, ..., n, 1) for n = 2, 3, 4, ... [i.e., a(2) = (2, 1)*(2, 1), a(3) = (3, 2, 1)*(2, 3, 1)]. - Clark Kimberling
a(n) = a(n-1) + A034856(n+1) = A000297(n-1) + 1 = A000217(n) + A000292(n+1) = A000290(n-1) + A000292(n). - Henry Bottomley, Oct 25 2001
a(n) = Sum_{0<=k, l<=n; k+l|n} k*l. - Ralf Stephan, May 06 2005
G.f.: (1+x-x^2)/(1-x)^4. - Colin Barker, Jan 06 2012
a(n) = A000330(n+1) - A000292(n-1). - Bruce J. Nicholson, Jul 05 2018
E.g.f.: (6 +24*x +12*x^2 +x^3)*exp(x)/6. - G. C. Greubel, Sep 11 2019

A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).

Original entry on oeis.org

1, 11, 48, 141, 331, 672, 1232, 2094, 3357, 5137, 7568, 10803, 15015, 20398, 27168, 35564, 45849, 58311, 73264, 91049, 112035, 136620, 165232, 198330, 236405, 279981, 329616, 385903, 449471, 520986, 601152, 690712, 790449, 901187, 1023792, 1159173, 1308283

Offset: 0

N. J. A. Sloane

These are the conjectured numbers of d-dimensional partitions for n=6, coming from a formula proposed by MacMahon in the general case that turned out to be wrong. Still, here for n=6, MacMahon's formula coincides for d < 3 with the first three terms of A042984. - Michel Marcus, Aug 16 2013

Binomial transform of [1,10,27,29,12,1,0,0,0,...], 6th row of A116672. - R. J. Mathar, Jul 18 2017

G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007326, A007327, A042984, A116672.

List([0..40], n-> (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120); # G. C. Greubel, Sep 11 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+5*x-3*x^2-2*x^3)/(1-x)^6 )); // G. C. Greubel, Sep 11 2019

seq(1+10*n+27*binomial(n,2)+29*binomial(n,3)+12*binomial(n,4)+binomial(n,5), n=0..40);

Table[1+10n+27Binomial[n,2]+29Binomial[n,3]+12Binomial[n,4]+ Binomial[n,5], {n,0,40}] (* Harvey P. Dale, Jul 27 2011 *)
CoefficientList[Series[(1+5x-3x^2-2x^3)/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 17 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,48,141,331,672},40] (* Harvey P. Dale, Aug 28 2019 *)

my(x='x+O('x^40)); Vec((1+5*x-3*x^2-2*x^3)/(1-x)^6) \\ G. C. Greubel, Sep 11 2019

[(120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120 for n in (0..40)] # G. C. Greubel, Sep 11 2019

G.f.: (1 + 5*x - 3*x^2 - 2*x^3)/(1-x)^6. - Colin Barker, Sep 05 2012
From G. C. Greubel, Sep 11 2019: (Start)
a(n) = (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120.
E.g.f.: (120 + 1200*x + 1620*x^2 + 580*x^3 + 60*x^4 + x^5)*exp(x)/120. (End)

Description corrected by Alford Arnold, Aug 1998

More terms added by G. C. Greubel, Sep 11 2019

A008779 Number of n-dimensional partitions of 5.

Original entry on oeis.org

1, 7, 24, 59, 120, 216, 357, 554, 819, 1165, 1606, 2157, 2834, 3654, 4635, 5796, 7157, 8739, 10564, 12655, 15036, 17732, 20769, 24174, 27975, 32201, 36882, 42049, 47734, 53970, 60791, 68232, 76329, 85119, 94640, 104931, 116032, 127984, 140829, 154610, 169371

Offset: 0

N. J. A. Sloane

a(n) = number of (n+8)-bit binary sequences with exactly 8 1's none of which is isolated. - David Callan, Jul 15 2004
For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
Binomial transform of [1,6,11,7,1,0,0,0,...], the 5th row of A116672. - R. J. Mathar, Jul 18 2017

G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.

G. C. Greubel, Table of n, a(n) for n = 0..1000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A116672, A289656.

List([0..45], n-> (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24); # G. C. Greubel, Sep 11 2019

[(n+1)*(n^3+21*n^2+38*n+24)/24: n in [0..45]]; // Vincenzo Librandi, May 21 2015

I:=[1,7,24,59,120]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, May 21 2015

seq(1+6*n+11*binomial(n,2)+7*binomial(n,3)+binomial(n,4), n=0..45);

CoefficientList[Series[(1+2*x-x^2-x^3)/(1-x)^5, {x,0,45}], x] (* Vincenzo Librandi, May 21 2015 *)
LinearRecurrence[{5,-10,10,-5,1}, {1,7,24,59,120}, 46] (* G. C. Greubel, Sep 11 2019 *)

Vec((-1+x^3+x^2-2*x)/(x-1)^5 + O(x^45)) \\ Altug Alkan, Jan 07 2016

[(n+1)*(n^3 + 21*n^2 + 38*n + 24)/24 for n in (0..45)] # G. C. Greubel, Sep 11 2019

G.f.: (1 +2*x -x^2 -x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24. - M. F. Hasler, Sep 15 2009
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). - Vincenzo Librandi, May 21 2015
E.g.f.: (24 + 144*x + 132*x^2 + 28*x^3 + x^4)*exp(x)/24. - G. C. Greubel, Sep 11 2019

More terms from Vincenzo Librandi, May 21 2015

cp's OEIS Frontend

A116673 Row sums of triangle A116672, in which the binomial transform of the n-th row lists the Euler transform of the n-th sequence in A007318 (Pascal's Triangle).

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A289656 Triangle read by rows: row n gives the first n terms of the binomial transform of the n-th row of A116672.

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A008778 a(n) = (n+1)*(n^2 +8*n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.

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A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).

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GAP

Magma

Maple

Mathematica

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Sage

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A008779 Number of n-dimensional partitions of 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A008778 a(n) = (n+1)(n^2 +8n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.