A092364 -id:A092364 - cp's OEIS frontend

A006002 a(n) = n*(n+1)^2/2.

0, 2, 9, 24, 50, 90, 147, 224, 324, 450, 605, 792, 1014, 1274, 1575, 1920, 2312, 2754, 3249, 3800, 4410, 5082, 5819, 6624, 7500, 8450, 9477, 10584, 11774, 13050, 14415, 15872, 17424, 19074, 20825, 22680, 24642, 26714, 28899, 31200, 33620, 36162, 38829, 41624

Offset: 0

N. J. A. Sloane

a(n) is the largest number that is not the sum of distinct numbers of form kn+1, k >= 0. - David W. Wilson, Dec 11 1999
Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ... Sum of the terms in brackets. Or sum of n consecutive integers beginning with T(n) + 1, where T(n) = n(n+1)/2. - Amarnath Murthy, Aug 27 2005
Apparently this is also the splittance (as defined by Hammer & Simeone, 1977) of the Kneser graphs of the form K(n+3,2). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 13 2009
Row sums of triangle A159797. - Omar E. Pol, Jul 24 2009
The same results occur when one plots the points (1,3), (3,6), (6,10), (10,15), and so on, for all the triangular numbers and finds the area beneath. Take three consecutive triangular numbers and label them a, b, c; the area created is simply (b-a)*(b+c)/2. Thus for 6,10,15 the area beneath the line defined by the points (6,10) and (10,15) is (10-6)*(10+15)/2 = 50. - J. M. Bergot, Jun 28 2011
Let P = ab where a and b are nonequal prime numbers > 1. Let Q be the product of all divisors of P^n. Q can be expressed as P^k, where k = n*(n+1)^2/2. This follows from the fact that all divisors are of the form a^i*b^j, for i,j from 0 to n. An example is given below. In the more general case, where P is the product of m nonequal prime numbers, k = n*(n+1)^m/2. When m=3, the sequence is the same as A092364. - James A. Raymond & Douglas Raymond, Dec 04 2011
For n > 0: sum of n-th row in A014132, seen as a triangle read by rows. - Reinhard Zumkeller, Dec 12 2012
Partial sums of A005449. - Omar E. Pol, Jan 12 2013
a(n) is the sum of x (or y) coordinates for an n X n square lattice in the upper right quadrant of Z^2 whose corner points are (0, 0), (0, n), (n, 0), and (n, n). - Joseph Wheat, Feb 03 2018
a(n) is the number of permutations of [n+2] that contain exactly 2 elements which are not left-to-right minimal. E.g., the 9 permutations comprising a(2) are 2134, 2143, 3124, 3142, 4123, 4132, 2314, 2413, 3412. - Andy Niedermaier, May 07 2022

			Let P^n=6^2. The product of the divisors of 36 = 10077796 = 6^9, i.e., for n=2, k=9. - _James A. Raymond_ & Douglas Raymond, Dec 04 2011

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 = 0..1000
Quang T. Bach, Roshil Paudyal and Jeffrey B. Remmel, A Fibonacci analogue of Stirling numbers, arXiv preprint arXiv:1510.04310 [math.CO], 2015.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Dexter Jane L. Indong and Gilbert R. Peralta, Inversions of permutations in Symmetric, Alternating, and Dihedral Gropus, JIS, Vol. 11 (2008), Article 08.4.3.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for two-way infinite sequences.

Cf. A002411: -a(-1-n).
Cf. A000217, A000292, A000330, A014132, A035006, A045991, A092364, A159797, A163274.
Cf. A000914 (partial sums), A005449 (first differences).
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Cf. A002378, A027480.
A bisection of A330298.

List([0..10^3],n->n*(n+1)^2/2); # Muniru A Asiru, Feb 04 2018

a006002 n = n * (n + 1) ^ 2 `div` 2  -- Reinhard Zumkeller, Dec 12 2012

[n*(n+1)^2/2 : n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2014

seq(binomial(n+1,2)*(n+1), n=0..36); # Zerinvary Lajos, Apr 25 2007

Table[(n^3-n^2)/2, {n, 41}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1}, {0,2,9,24}, 40] (* Harvey P. Dale, Aug 14 2012 *)
Accumulate @ # (# + 1) & [Range[0,50]] (* Waldemar Puszkarz, Jan 24 2015 *)

PARI
```
a(n)=n*(n+1)^2/2
```

G.f.: x*(x + 2)/(1 - x)^4. - Michael Somos, Jan 30 2004
a(n) = (n + 1) * binomial(n+1, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A035006(n+1)/4. - Johannes W. Meijer, Feb 04 2010
a(n) = 2*binomial(n+1, 2) + 3*binomial(n+1, 3). - Gary Detlefs, Jun 06 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 14 2012
a(n) = A000292(n) + A000330(n). - Omar E. Pol, Jan 11 2013
a(n) = A045991(n+1)/2. - J. M. Bergot, Aug 10 2013
a(n) = Sum_{j=1..n} Sum_{i=1..j} (2*j - i + 1). - Wesley Ivan Hurt, Nov 17 2014
a(n) = Sum_{i=0..n} n*(n - i) + i. - Bruno Berselli, Jan 13 2016
a(n) = t(n, A000217(n)), where t(h,k) = A000217(h) + h*k. - Bruno Berselli, Feb 28 2017
Sum_{n>0} 1/a(n) = 4 - Pi^2/3. - Jaume Oliver Lafont, Jul 11 2017 [corrected by Amiram Eldar, Jan 28 2022]
E.g.f.: exp(x)*x*(4 + 5*x + x^2)/2. - Stefano Spezia, May 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/6 + 4*log(2) - 4. - Amiram Eldar, Jan 28 2022
From J.S. Seneschal, Jun 27 2024: (Start)
a(n) = (A002378(n)^2/2)/n = (n+1)/2 * A002378(n).
a(n) = A027480(n) - A000217(n). (End)

A085537 a(n) = n^4 - n^3.

Original entry on oeis.org

0, 0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for a fixed x in {1,2,3,4} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Let K_n denote the complete graph on n (n>1) vertices. The sequence corresponds to the Wiener index of K_n X K_n (Cartesian product of K_n with itself). - K.V.Iyer, Mar 12 2009
Lewis proved that the order of a solvable nonabelian finite group |G| is less than or equal to e^4 - e^3, where when d is an irreducible character degree of G, then there is a positive integer e such that |G| = d(d+e). - Jonathan Vos Post, Jun 21 2012

Michael B. Porter, Table of n, a(n) for n = 0..100000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Mark L. Lewis, Bounding group orders by large character degrees: A question of Snyder, arXiv:1206.4334 [math.GR], Jun 19 2012.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A228273.
Cf. A000578, A000583, A092364, A146489.
Cf. A085540 (same sequence with initial 0 dropped).

Table[(n - 1) n^3, {n, 0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 54, 192, 500}, {0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[2 x^2 (4 + 7 x + x^2)/(1 - x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI
```
A085537(n) = n^4-n^3
```

From R. J. Mathar, Sep 12 2008: (Start)
a(n) = A085540(n-1).
G.f.: 2*x^2*(4 + 7*x + x^2)/(1-x)^5. (End)
a(n) = A000583(n) - A000578(n). - Omar E. Pol, Jun 23 2012
Sum_{n>=2} 1/a(n) = 3 - zeta(2) - zeta(3) = A152419. - Daniel Suteu, Feb 06 2017
a(n) = 2*A092364(n+1). - Bruno Berselli, Sep 08 2017
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. - Amiram Eldar, Jul 05 2020
E.g.f.: exp(x)*x^2*(4 + 5*x + x^2). - Stefano Spezia, Jul 06 2021
Product_{n>=2} (1 - 1/a(n)) = A146489. - Amiram Eldar, Nov 22 2022

A221857 Number A(n,k) of shapes of balanced k-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 3, 4, 1, 0, 1, 1, 5, 6, 1, 4, 1, 0, 1, 1, 6, 10, 4, 9, 4, 1, 0, 1, 1, 7, 15, 10, 1, 27, 1, 1, 0, 1, 1, 8, 21, 20, 5, 16, 27, 8, 1, 0, 1, 1, 9, 28, 35, 15, 1, 96, 81, 16, 1, 0, 1, 1, 10, 36, 56, 35, 6, 25, 256, 81, 32, 1, 0

Offset: 0

Alois P. Heinz, Apr 10 2013

			: A(2,2) = 2  : A(2,3) = 3      : A(3,3) = 3          :
:   o     o   :   o    o    o   :   o      o      o   :
:  / \   / \  :  /|\  /|\  /|\  :  /|\    /|\    /|\  :
: o         o : o      o      o : o o    o   o    o o :
:.............:.................:.....................:
: A(3,4) = 6                                          :
:    o        o        o        o       o        o    :
:  /( )\    /( )\    /( )\    /( )\   /( )\    /( )\  :
: o o      o   o    o     o    o o     o   o      o o :
Square array A(n,k) begins:
  1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...
  1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...
  0, 1, 2,  3,   4,   5,  6,  7,  8,   9,  10, ...
  0, 1, 1,  3,   6,  10, 15, 21, 28,  36,  45, ...
  0, 1, 4,  1,   4,  10, 20, 35, 56,  84, 120, ...
  0, 1, 4,  9,   1,   5, 15, 35, 70, 126, 210, ...
  0, 1, 4, 27,  16,   1,  6, 21, 56, 126, 252, ...
  0, 1, 1, 27,  96,  25,  1,  7, 28,  84, 210, ...
  0, 1, 8, 81, 256, 250, 36,  1,  8,  36, 120, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Jeffrey Barnett, Counting Balanced Tree Shapes, 2007
Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv:1107.3472 [math.CO], Apr 2012

Columns k=1-10 give: A000012, A110316, A131889, A131890, A131891, A131892, A131893, A229393, A229394, A229395.
Rows n=0+1, 2-3, give: A000012, A001477, A179865.
Diagonal and upper diagonals give: A028310, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288.
Lower diagonals give: A000012, A000290, A092364(n) for n>1.

A:= proc(n, k) option remember; local m, r; if n<2 or k=1 then 1
      elif k=0 then 0 else r:= iquo(n-1, k, 'm');
      binomial(k, m)*A(r+1, k)^m*A(r, k)^(k-m) fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n-1, k]; Binomial[k, m]*a[r+1, k]^m*a[r, k]^(k-m)]]]; Table[a[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 17 2013, translated from Maple *)

A019582 a(n) = n*(n - 1)^3/2.

Original entry on oeis.org

0, 0, 1, 12, 54, 160, 375, 756, 1372, 2304, 3645, 5500, 7986, 11232, 15379, 20580, 27000, 34816, 44217, 55404, 68590, 84000, 101871, 122452, 146004, 172800, 203125, 237276, 275562, 318304, 365835, 418500, 476656, 540672, 610929, 687820, 771750, 863136

Offset: 0

N. J. A. Sloane, Dec 11 1996

a(n) = n(n-1)^3/2 is half the number of colorings of 4 points on a line with n colors. - R. H. Hardin, Feb 23 2002
n^2*n(n+1)/2: a(n+1) = product of n-th triangular number and n-th square number. E.g., a(4)=6*9=54. - Alexandre Wajnberg, Dec 18 2005
Also, the number of ways to place two dominoes horizontally in different rows on an n X n chessboard. - Ralf Stephan, Jun 09 2014
a(n) is the second Zagreb index of the complete graph K[n]. The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g. - Emeric Deutsch, Nov 07 2016
a(n+1) is the number of inequivalent 2 X 2 matrices with entries in {1,2,3,...,n} when a matrix and its transpose are considered equivalent. - David Nacin, Feb 27 2017

			G.f. = x^2 + 12*x^3 + 54*x^4 + 160*x^5 + 375*x^6 + 756*x^7 + 1372*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..680
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000290, A092364.

A row or column of A132191.

[n*(n-1)^3/2: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f := n->n*(n-1)^3/2; seq(f(n), n=0..50);

f[n_]:=n*(n-1)^3/2; Table[f[n], {n,0,4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2010 *)

a(n)=n*(n-1)^3/2 \\ Charles R Greathouse IV, Feb 27 2017

a(n+1) = Sum_{k=0..n} n^2(n-k) = n^3*(n+1)/2. - Paul Barry, Sep 02 2003
a(n+1) = A000290(n) * A000217(n). - Zerinvary Lajos, Jan 20 2007
Sum_{j>=2} 1/a(j) = hypergeom([1, 1, 1, 1], [2, 2, 3], 1) = 2 - 2*zeta(2) + 2*zeta(3). - Stephen Crowley, Jun 28 2009
G.f.: -x^2*(4*x^2 + 7*x + 1)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
a(1 - n) = A092364(n). - Michael Somos, Jun 09 2014
Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/2 - zeta(2) + 4*log(2) - 2. - Amiram Eldar, Sep 11 2022
E.g.f.: exp(x)*x^2*(1 + 3*x + x^2)/2. - Stefano Spezia, Jun 10 2023

A085540 a(n) = n*(n + 1)^3.

Original entry on oeis.org

0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

N. J. A. Sloane, Jul 05 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A085537 (same sequence with a 0 prepended), A092364.

[n*(n+1)^3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

Table[n (n + 1)^3, {n, 0, 40}] (* Vincenzo Librandi, Aug 15 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{0,8,54,192,500},40] (* Harvey P. Dale, May 06 2019 *)

a(n) = 2*A092364(n+1). - Zerinvary Lajos, May 09 2007
G.f.: -2*x*(4 + 7*x + x^2)/(x - 1)^5. - R. J. Mathar, Mar 10 2011
a(n) = A085537(n-1). - Eric W. Weisstein, Sep 08 2017
E.g.f.: exp(x)*x*(8 + 19*x + 9*x^2 + x^3). - Stefano Spezia, Jun 10 2023
From Amiram Eldar, Jul 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi^2/6 - zeta(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. (End)

A239592 a(n) = (n^4 - n^3 + 4*n^2 + 2)/2.

Original entry on oeis.org

1, 3, 13, 46, 129, 301, 613, 1128, 1921, 3079, 4701, 6898, 9793, 13521, 18229, 24076, 31233, 39883, 50221, 62454, 76801, 93493, 112773, 134896, 160129, 188751, 221053, 257338, 297921, 343129, 393301, 448788, 509953, 577171, 650829, 731326, 819073, 914493

Offset: 0

Philippe Deléham, Mar 22 2014

Main diagonal of square array A239331.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A239331.

[(n^4-n^3+4*n^2 + 2)/2: n in [0..40]]; // Vincenzo Librandi, Mar 23 2014

CoefficientList[Series[(1 - 2 x + 8 x^2 + x^3 + 4 x^4)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)

Vec((1-2*x+8*x^2+x^3+4*x^4)/(1-x)^5 + O(x^100)) \\ Colin Barker, Mar 22 2014

G.f.: (1 - 2*x + 8*x^2 + x^3 + 4*x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), a(0) = 1, a(1) = 3, a(2) = 13, a(3) = 46, a(4) = 129.
a(n) = A058331(n) + A092364(n).

A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 27, 24, 1, 16, 96, 208, 128, 1, 25, 250, 950, 1325, 610, 1, 36, 540, 3120, 7290, 7416, 2742, 1, 49, 1029, 8330, 28665, 47922, 38563, 11970, 1, 64, 1792, 19264, 90720, 219968, 287168, 191808, 51424, 1, 81, 2916, 40068, 246078, 806274, 1509732, 1619676, 926073, 218718

Offset: 0

R. J. Mathar, Sep 02 2017

			1 ;
1 1;
1 4 4;
1 9 27 24;
1 16 96 208 128;
1 25 250 950 1325 610;
1 36 540 3120 7290 7416 2742;
1 49 1029 8330 28665 47922 38563 11970;
1 64 1792 19264 90720 219968 287168 191808 51424;
1 81 2916 40068 246078 806274 1509732 1619676 926073 218718;

A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5, Corollary 9, Table 2.1

Cf. A092364 (column r=2), A289713 (row sums)

A289710 := proc(n,r)
    if r = 0 then
        1;
    else
        r*binomial(n,r)*binomial(n+r-1,n-1)-n*(r-1)*binomial(n,r) ;
    end if ;
end proc:

cp's OEIS Frontend

A006002 a(n) = n*(n+1)^2/2.

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A085537 a(n) = n^4 - n^3.

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A221857 Number A(n,k) of shapes of balanced k-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A019582 a(n) = n*(n - 1)^3/2.

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A085540 a(n) = n*(n + 1)^3.

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A239592 a(n) = (n^4 - n^3 + 4*n^2 + 2)/2.

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Magma

Mathematica

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A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.

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