A157705 -id:A157705 - cp's OEIS frontend

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

1, 7, 24, 58, 115, 201, 322, 484, 693, 955, 1276, 1662, 2119, 2653, 3270, 3976, 4777, 5679, 6688, 7810, 9051, 10417, 11914, 13548, 15325, 17251, 19332, 21574, 23983, 26565, 29326, 32272, 35409, 38743, 42280, 46026, 49987, 54169, 58578, 63220

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009
Row 1 of the convolution arrays A213831 and A213833. - Clark Kimberling, Jul 04 2012
Partial sums of A056109. - J. M. Bergot, Jun 22 2013
Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014
Row sums of A244418. - L. Edson Jeffery, Jan 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Christian Barrientos, The number of spanning trees of cyclic snakes, Indones. J. Comb. (2025) Vol. 9, No. 1, 21-30. See p. 29.
J. A. Dias da Silva and Pedro J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081435, A081437, A156933, A157705, A244418.

List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

A081436 := proc(n)
    (n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n),n=0..60) ; # R. J. Mathar, Jun 26 2013

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013

[(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.
G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)
a(n) = A110449(n, n-1), for n>1.
a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021

G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009

A156933 FP4 polynomials related to the o.g.f.s of the columns of the A156925 matrix.

Original entry on oeis.org

1, 1, 1, -11, 156, -627, 736, 591, -1116, -369, -6, 106, -2772, 76070, -1087552, 8632650, -40358780, 106452214, -99774996, -284430514, 1125952500, -1581820542, 737716032, 414532350, -357790500, -81870750, -1275750

Offset: 0

Johannes W. Meijer, Feb 20 2009

For the matrix of the coefficients of the FP2 see A156925. The coefficients in the columns of this matrix are the powers of z^m, m=0, 1, 2, ... . The columns are numbered: 1, 2, 3, ... .
The GF4(z;m) generate the sequences of the z^m coefficients. The general structure of the GF4(z;m) is given below.
The FP4(z,m) in the numerator of the GF4(z;m) is a polynomial of a certain degree, let's say k4. The (k4+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP4(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF2 formulas, see A156925.
An appropriate name for the polynomials FP4(z;m) in the numerators of the GF4(z;m) seems to be the flower polynomials of the fourth kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP4 and the FP3, see A156927, resemble each other closely and look like the zero patterns of the FP1 and FP2.
The sequence of (k4+1) number of terms of the FP4(z;m) polynomials for m from 0 to 11 is 1, 2, 7, 17, 28, 44, 63, 83, 108, 136, 167, 199.

			The first few rows of the "triangle" of the FP4(z;m) coefficients are:
[1]
[1, 1]
[ -11, 156, -627, 736, 591, -1116, -369]
The first few FP4 polynomials are:
FP4(z; m=0) = 1
FP4(z; m=1) = (1+z)
FP4(z; m=2) = ( -11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6 )
Some GF4(z;m) are:
GF4(z;m=1) = z*(1+z)/((1-3*z)*(1-z)^4)
GF4(z;m=2) = z^2*(-11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6)/((1-z)^7*(1-3*z)^4*(1-5*z))

Cf. A156920, A156921, A156925, A156927.
For the first few GF4's see A156934, A156935, A156936, A156937.
Row sums A156938.
For the polynomials in the denominators of the GF4(z;m) see A157705. - Johannes W. Meijer, Mar 07 2009

G.f.: GF4(z;m):= z^q*FP4(z;m) / Product_{k=0..m} (1-(2*m+1-(2*k))*z)^(3*k+1).

A064350 a(n) = (3*n)!/n!.

Original entry on oeis.org

1, 6, 360, 60480, 19958400, 10897286400, 8892185702400, 10137091700736000, 15388105201717248000, 30006805143348633600000, 73096577329197271449600000, 217535414131691079834009600000, 776601428450137155007414272000000, 3275704825202678519821273399296000000

Offset: 0

Karol A. Penson, Sep 18 2001

Also a(n) = (((n)!)^2)*A006480(n). [corrected by Johannes W. Meijer, Mar 02 2009]

a(n) is the number of ways to partition the set {1,2,...,3n} into n blocks of size 3 and then linearly order the elements within each block. - Geoffrey Critzer, Dec 30 2012

Harry J. Smith, Table of n, a(n) for n = 0..70
Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; arXiv preprint, arXiv:quant-ph/0111151, 2001.

Cf. A006480, A248759.
From Johannes W. Meijer, Mar 07 2009: (Start)
Equals A001525*3!
Equals row sums of A157704 and A157705. (End)

Table[(3n)!/n!,{n,0,20}]  (* Geoffrey Critzer, Dec 30 2012 *)

{ t=f=1; for (n=0, 70, if (n, t*=3*n*(3*n - 1)*(3*n - 2); f*=n); write("b064350.txt", n, " ", t/f) ) } \\ Harry J. Smith, Sep 12 2009

Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*BesselK(1/3, 2*sqrt(x/27))/(3*Pi*sqrt(x)) dx, n >= 0.
A recursive formula: a(n) = (27 * (n - 1)^2 + 27 * (n - 1) + 6) * a(n - 1) with a(0) = 1. An explicit formula following from the recursion equation: a(n) = (3/2)*27^n*GAMMA(n+2/3)*GAMMA(n+1/3)/(Pi*3^(1/2)). - Thomas Wieder, Nov 15 2004
E.g.f.: (of aerated sequence) 2*cos(arcsin((3*sqrt(3)*x/2)/3))/sqrt(4-27*x^2). - Paul Barry, Jul 27 2010
E.g.f.: (with interpolated zeros) exp(x^3). - Geoffrey Critzer, Dec 30 2012
Sum_{n>=1} 1/a(n) = A248759. - Amiram Eldar, Nov 15 2020

a(11) from Harry J. Smith, Sep 12 2009

A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

Original entry on oeis.org

1, 1, 5, 32, 186, 132, 10, 56, 2814, 17834, 27324, 11364, 1078, 10, 48, 17988, 494720, 3324209, 7526484, 6382271, 2004296, 203799, 4580, 5, 16, 72210, 7108338, 146595355, 1025458635, 2957655028, 3828236468

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF3(z;n) polynomials in the GF3 denominators of A156927 can be found below.
The general structure of the GFKT3(z;p) that generate the z^p coefficients of the PDGF3(z; n) polynomials can also be found below. The KT3(z;p) polynomials in the numerators of the GFKT3(z; p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT3(z;p) polynomials is 1, 2, 4, 7, 10, 13, 14, 17, 20, 23, 26, 29, 32, 34, 36, 39, 42. The differences of this sequence and that of the number of terms of the KT4(z;p), see A157705, follow a simple pattern.
A Maple algorithm that generates relevant GFKT3(z;p) information can be found below.

			Some PDGF3 (z;n) are:
  PDGF3(z;n=3) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10
  PDGF3(z;n=4) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10*(1-5*z)^13
The first few GFKT3's are:
  GFKT3(z;p=0) = 1/(1-z)
  GFKT3(z;p=1) = -(5*z+1)/(1-z)^4
  GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1-z)^7
Some KT3(z,p) polynomials are:
  KT3(z;p=2) = 32+186*z+132*z^2+10*z^3
  KT3(z;p=3) = 56+2814*z+17834*z^2+27324*z^3+11364*z^4+1078*z^5+10*z^6

Originator sequence A156927.
See A002414 for the z^1 coefficients and A157707 for the z^2 coefficients divided by 2.
Row sums equal A064350 and those of A157705.
Cf. A157702, A157703, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(k+1)*z)^(1+3*k), k=0..n2): a(n2):= coeff(fz(n2),z,p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT3(p):=sum((fk)*z^k, k=0..infinity); q3:=ldegree((numer(GFKT3(p)))): KT3(p):=sort((-1)^(p)*simplify((GFKT3(p)*(1-z)^(3*p+1))/z^q3),z, ascending);

PDGF3(z;n) = Product_{k=0..n} (1-(k+1)*z)^(1+3*k) with n = 1, 2, 3, ...
GFKT3(z;p) = (-1)^(p)*(z^q3)*KT3(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A157702 G.f.s of the z^p coefficients of the polynomials in the GF1 denominators of A156921.

Original entry on oeis.org

1, 1, 1, 7, 26, 7, 3, 166, 951, 951, 166, 3, 263, 8999, 59637, 108602, 59637, 8999, 263, 174, 33124, 848555, 6062651, 15477896, 15477896, 6062651, 848555, 33124, 174, 45, 66963, 5856626, 122966782, 920090513

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF1(z;n) polynomials in the GF1 denominators of A156921 can be found below.
The general structure of the GFKT1(z;p) that generate the z^p coefficients of the PDGF1(z; n) polynomials can also be found below. The KT1(z;p) polynomials in the numerators of the GFKT1(z;p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT1(z;p) polynomials is: 1, 2, 3, 6, 7, 10, 13, 14, 17, 20, 23, 24, 27, 30, 33, 36, 37, 40. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124.
A Maple algorithm that generates relevant GFKT1(z;p) information can be found below.

			Some PDGF1 (z;n) are:
  PDGF1(z;n=3) = (1-5*z)*(1-3*z)^2*(1-z)^3
  PDGF1(z;n=4) = ((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)
The first few GFKT1's are:
  GFKT1(z;p=0) = 1/(1-z)
  GFKT1(z;p=1) = -z*(1+z)/(1-z)^4
  GFKT1(z;p=2) = z^2*(7+26*z+7*z^2)/(1-z)^7
Some KT1(z;p) polynomials are:
  KT1(z;p=2) = 7+26*z+7*z^2
  KT1(z;p=3) = 3+166*z+951*z^2+951*z^3+166*z^4+3*z^5
  KT1(z;p=4) = 263+8999*z+59637*z^2+108602*z^3+59637*z^4+8999*z^5+263*z^6

Originator sequence A156921.
See A000330 for the z^1 coefficients and A157706 for the z^2 coefficients.
Row sums equal A052502.
Cf. A157703, A157704, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn, a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(2*m-1)*z)^(n2+1-m),m=1..n2); a(n2):= coeff(fz(n2),z,p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT1(p):=(sum(fk*z^k,k=0..infinity)); q1:=ldegree((numer(GFKT1(p)))): KT1(p):=sort((-1)^p*simplify((GFKT1(p))*(1-z)^(3*p+1)/z^q1),z, ascending);

PDGF1(z;n) = Product_{m=1..n} (1-(2*m-1)*z)^(n+1-m) with n = 1, 2, 3, ... .
GFKT1(z;p) = (-1)^(p)*(z^q1)*KT1(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ... .
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A157703 G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.

Original entry on oeis.org

1, 1, 5, 5, 2, 62, 152, 62, 2, 91, 1652, 5957, 5957, 1652, 91, 52, 5240, 77630, 342188, 551180, 342188, 77630, 5240, 52, 12, 8549, 424921, 5629615, 28123559, 61108544, 61108544, 28123559, 5629615, 424921, 8549, 12

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF2(z;n) polynomials in the GF2 denominators of A156925 can be found below.
The general structure of the GFKT2(z;p) that generate the z^p coefficients of the PDGF2(z; n) polynomials can also be found below. The KT2(z;p) polynomials in the numerators of the GFKT2(z;p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT2(z;p) polynomials is: 1, 1, 2, 5, 6, 9, 12, 13, 16, 19, 22, 23, 26. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124 with one exception, here a(0) = 0.
A Maple algorithm that generates relevant GFKT2(z;p) information can be found below.

			Some PDGF2 (z;n) are:
  PDGF2(z;n=3) = (1-z)^3*(1-2*z)^2*(1-3*z)
  PDGF2(z;n=4) = (1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z)
The first few GFKT2's are:
  GFKT2(z;p=0) = 1/(1-z)
  GFKT2(z;p=1) = -z/(z-1)^4
  GFKT2(z;p=2) = z^2*(5+5*z)/(1-z)^7
Some KT2(z,p) polynomials are:
  KT2(z;p=2) = 5+5*z
  KT2(z;p=3) = 2+62*z+152*z^2+62*z^3+2*z^4
  KT2(z;p=4) = 91+1652*z+5957*z^2+5957*z^3+1652*z^4+91*z^5

Originator sequence A156925.
See A000292 for the z^1 coefficients and A040977 for the z^2 coefficients divided by 5.
Row sums equal A025035.
Cf. A157702, A157704, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-m*z)^(n2+1-m),m=1..n2): a(n2):= coeff(fz(n2),z,p): end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT2(p):=sum((fk)*z^k,k=0..infinity); q2:=ldegree((numer(GFKT2(p)))): KT2(p):=sort((-1)^p*simplify((GFKT2(p)*(1-z)^(3*p+1))/z^q2),z, ascending);

PDGF2(z;n) = Product_{m=1..n} (1-m*z)^(n+1-m) with n = 1, 2, 3, ...
GFKT2(z;p) = (-1)^(p)*(z^q2)*KT2(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A001525 a(n) = (3n)!/(3!n!).

Original entry on oeis.org

1, 60, 10080, 3326400, 1816214400, 1482030950400, 1689515283456000, 2564684200286208000, 5001134190558105600000, 12182762888199545241600000, 36255902355281846639001600000, 129433571408356192501235712000000, 545950804200446419970212233216000000

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100

Equals A064350/3!, row sums of A157704/3! and row sums of A157705/3!.

Table[(3 n)!/(3! n!), {n, 20}] (* Harvey P. Dale, May 25 2011 *)

More terms from Harvey P. Dale, May 25 2011

A157708 The z^2 coefficients of the polynomials in the GF4 denominators of A156933.

Original entry on oeis.org

18, 254, 1571, 6335, 19615, 50743, 115234, 237066, 451320, 807180, 1371293, 2231489, 3500861, 5322205, 7872820, 11369668, 16074894, 22301706, 30420615, 40866035, 54143243, 70835699, 91612726

Offset: 1

Johannes W. Meijer, Mar 07 2009

See A157705 for background information.

Cf. A156933, A157705

nmax:=23; for n from 0 to nmax do fz(n):=product((1-(2*n+1-2*k)*z)^(3*k+1), k=0..n); c(n):= coeff(fz(n),z,2); end do: a:=n-> c(n): seq(a(n), n=1..nmax);

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7)
a(n) = 1/2*n^6+5/2*n^5+41/8*n^4+67/12*n^3+27/8*n^2+11/12*n
G.f.: (18 + 128*z + 171*z^2 + 42*z^3 + z^4)/(1-z)^7

cp's OEIS Frontend

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

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A156933 FP4 polynomials related to the o.g.f.s of the columns of the A156925 matrix.

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A064350 a(n) = (3*n)!/n!.

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A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

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A157702 G.f.s of the z^p coefficients of the polynomials in the GF1 denominators of A156921.

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A157703 G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.

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A001525 a(n) = (3n)!/(3!n!).

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A157708 The z^2 coefficients of the polynomials in the GF4 denominators of A156933.

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