A002412 -id:A002412 - cp's OEIS frontend

A093561 (4,1) Pascal triangle.

1, 4, 1, 4, 5, 1, 4, 9, 6, 1, 4, 13, 15, 7, 1, 4, 17, 28, 22, 8, 1, 4, 21, 45, 50, 30, 9, 1, 4, 25, 66, 95, 80, 39, 10, 1, 4, 29, 91, 161, 175, 119, 49, 11, 1, 4, 33, 120, 252, 336, 294, 168, 60, 12, 1, 4, 37, 153, 372, 588, 630, 462, 228, 72, 13, 1, 4, 41, 190, 525, 960, 1218

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(4;n,m) gives in the columns m >= 1 the figurate numbers based on A016813, including the hexagonal numbers A000384 (see the W. Lang link).
This is the fourth member, d=4, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 and A093560, for d=1..3.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+3*z)/(1-(1+x)*z).
The SW-NE diagonals give A000285(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018

			Triangle begins
  [1];
  [4, 1];
  [4, 5, 1];
  [4, 9, 6, 1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, >Rows n = 0..125 of triangle, flattened
P. Bala, A note on the diagonals of a proper Riordan Array
W. Lang, First 10 rows and array of figurate numbers .

Cf. Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 3 for n=2 and 0 otherwise.
Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, A051947, A050483, A052181, A055843.
Cf. A007318, A093562 (d=5), A228196, A228576.

a093561 n k = a093561_tabl !! n !! k
a093561_row n = a093561_tabl !! n
a093561_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [4, 1]
-- Reinhard Zumkeller, Aug 31 2014

from math import comb, isqrt
def A093561(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+3*(r-a))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m) = F(4;n-m, m) for 0<= m <= n, otherwise 0, with F(4;0, 0)=1, F(4;n, 0)=4 if n>=1 and F(4;n, m) = (4*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 3*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 9*x + 6*x^2/2! + x^3/3!) = 4 + 13*x + 28*x^2/2! + 50*x^3/3! + 80*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 50, 25, 6, 22, 86, 130, 95, 36, 7, 29, 148, 296, 295, 161, 49, 8, 37, 239, 610, 791, 581, 252, 64, 9, 46, 367, 1163, 1897, 1792, 1036, 372, 81, 10, 56, 541, 2083, 4166, 4900, 3612, 1716, 525, 100, 11, 67, 771, 3544, 8518, 12174, 11088, 6672, 2685, 715, 121, 12

Offset: 1

R. H. Hardin, Apr 23 2013

Table starts
..2...4...7...11....16.....22.....29......37......46.......56.......67
..3...9..22...46....86....148....239.....367.....541......771.....1068
..4..16..50..130...296....610...1163....2083....3544.....5776.....9076
..5..25..95..295...791...1897...4166....8518...16414....30086....52834
..6..36.161..581..1792...4900..12174...27966...60172...122464...237590
..7..49.252.1036..3612..11088..30738...78354..186142...416394...884236
..8..64.372.1716..6672..22716..69498..194634..505912..1233584..2845492
..9..81.525.2685.11517..43065.144111..439791.1241383..3276559..8157227
.10.100.715.4015.18832..76714.278707..920491.2803658..7963384.21280337
.11.121.946.5786.29458.129844.508937.1808521.5911763.17978389.51325352
From Charles A. Lane, Aug 22 2013: (Start)
The first column is also the coefficients of a in y''[x] - a*x^n*y[x] + b*en*y[x] = 0 where n = 0. The recursion yields coefficients of a, a*b*en, a*b^2*en^2 etc.
The second column is obtained when n=1, the third column when n=2. The final column is for n=10.
Example: Write a normal recursion for n=4. For convenience set x to 1. Running the recursion yields
1-(b en)/2+(b^2 en^2)/24+1/30 (a-(b^3 en^3)/24)+(-384 a b en+b^4 en^4)/40320+(2064 a b^2 en^2-b^5 en^5)/3628800+(120960 a^2-7104 a b^3 en^3+b^6 en^6)/479001600+(-4682880 a^2 b en+18984 a b^4 en^4-b^7 en^7)/87178291200+(54268416 a^2 b^2 en^2-43008 a b^5 en^5+b^8 en^8)/20922789888000.
The coefficient of a is 24, the coefficient of a b en is 384 and the coefficient of a b^2 en^2 is 2064. Dividing by 4! yields a sequence of 1,16,86... , the same as column 5 without the leading 1. There is a hint of unity among the oscillators. (End)

			Some solutions for n=3 k=4
..0..0..0..0....0..1..0..0....0..0..0..0....1..1..1..1....0..0..0..0
..0..0..0..0....0..1..1..0....0..0..0..0....1..1..1..1....1..1..0..0
..0..0..0..1....1..1..1..0....1..1..0..0....1..1..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..5304

Column 2 is A000290(n+1).
Column 3 is A002412(n+1).
Column 4 is A006324(n+1).
Row 1 is A000124.
Row 2 is A223718.
Row 3 is A223659.
Cf. A071920, A071921 (larger and reflected versions of table). - Alois P. Heinz, Sep 22 2013

T:= (n, k)-> add(binomial(k+2*j-1, 2*j), j=0..n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 22 2013

T[n_, k_] := Sum[Binomial[k + 2*j - 1, 2*j], {j, 0, n}]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

Empirical: columns k=1..7 are polynomials of degree k.
Empirical: rows n=1..7 are polynomials of degree 2n.
T(n,k) = Sum_{j=0..n} C(k+2*j-1,2*j). - Alois P. Heinz, Sep 22 2013

A228461 Two-dimensional array read by antidiagonals: T(n,k) = number of arrays of maxima of three adjacent elements of some length n+2 0..k array.

Original entry on oeis.org

2, 3, 4, 4, 9, 7, 5, 16, 22, 11, 6, 25, 50, 46, 17, 7, 36, 95, 130, 91, 27, 8, 49, 161, 295, 310, 183, 44, 9, 64, 252, 581, 821, 736, 383, 72, 10, 81, 372, 1036, 1847, 2227, 1821, 819, 117, 11, 100, 525, 1716, 3703, 5615, 6254, 4673, 1749, 189, 12, 121, 715, 2685, 6812

Offset: 1

R. H. Hardin Aug 22 2013

There are two arrays (or lists, or vectors) involved, a length n+2 array with free elements from 0..k (thus (k+1)^(n+2) of them) and an array that is being enumerated of length n, each element of the latter being the maximum of three adjacent elements of the first array.

Many different first arrays can give the same second array.

			Table starts
...2....3.....4.....5......6......7.......8.......9......10.......11.......12
...4....9....16....25.....36.....49......64......81.....100......121......144
...7...22....50....95....161....252.....372.....525.....715......946.....1222
..11...46...130...295....581...1036....1716....2685....4015.....5786.....8086
..17...91...310...821...1847...3703....6812...11721...19117....29843....44914
..27..183...736..2227...5615..12453...25096...46941...82699...138699...223224
..44..383..1821..6254..17487..42386...92430..185727..349558...623513..1063283
..72..819..4673.18394..57303.151882..357510..768231.1535578..2893605..5191407
.117.1749.12107.55285.194064.567835.1453506.3357985.7152815.14263777.26930773
Some solutions for n=4 k=4
..3....4....4....3....3....4....3....4....3....0....3....3....4....2....0....2
..0....4....1....3....2....0....2....4....1....0....3....3....1....2....0....0
..4....1....1....0....1....0....4....0....0....0....2....1....4....2....2....3
..4....0....3....3....2....0....4....2....3....1....3....1....4....0....4....3

R. H. Hardin, Table of n, a(n) for n = 1..1700

Column 1 is A005252(n+3)
Column 2 is A217878
Column 3 is A217949.
A228464 is another column.
Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Row 3 is A002412(n+1)
Row 4 is A006324(n+1)
See A217883, A217954 for similar arrays.

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-4)
k=2: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-4) -a(n-5) +a(n-6) +a(n-7)
k=3: [order 10]
k=4: [order 13]
k=5: [order 16]
k=6: [order 19]
k=7: [order 22]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = (2/3)*n^3 + (5/2)*n^2 + (17/6)*n + 1
n=4: a(n) = (1/3)*n^4 + 2*n^3 + (25/6)*n^2 + (7/2)*n + 1
n=5: a(n) = (2/15)*n^5 + (7/6)*n^4 + (25/6)*n^3 + (19/3)*n^2 + (21/5)*n + 1
n=6: [polynomial of degree 6]
n=7: [polynomial of degree 7]

Edited by N. J. A. Sloane, Sep 02 2013

A200886 T(n,k) is the number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.

Original entry on oeis.org

7, 22, 12, 50, 51, 21, 95, 144, 121, 37, 161, 325, 422, 292, 65, 252, 636, 1121, 1268, 704, 114, 372, 1127, 2507, 3985, 3823, 1691, 200, 525, 1856, 4977, 10213, 14288, 11472, 4059, 351, 715, 2889, 9052, 22736, 42182, 50995, 34350, 9749, 616, 946, 4300, 15393

Offset: 1

R. H. Hardin, Nov 23 2011

T(n,k) is the number of lattice points in k*P where P is a polytope of dimension n+2 in R^(n+2) whose vertices are lattice points, and therefore for each n it is an Ehrhart polynomial of degree n+2. This confirms the empirical formulas for the rows. - Robert Israel, Mar 21 2021

			Some solutions for n=4, k=3:
  1   2   3   0   0   1   2   3   0   1   2   3   3   1   2   2
  1   2   1   0   1   0   1   0   3   0   2   2   3   0   3   2
  2   2   3   0   2   2   3   2   3   0   3   3   3   1   3   0
  2   0   3   0   3   3   3   3   2   0   3   3   3   1   0   2
  1   1   2   1   3   3   2   3   0   1   3   3   3   1   2   3
  0   2   2   1   3   2   1   0   2   1   2   1   1   3   3   3
Table starts:
....7....22.....50......95......161.......252.......372........525........715
...12....51....144.....325......636......1127......1856.......2889.......4300
...21...121....422....1121.....2507......4977......9052......15393......24817
...37...292...1268....3985....10213.....22736.....45648......84681.....147565
...65...704...3823...14288....42182....105813....235538.....478467.....904111
..114..1691..11472...50995...173606....491533...1215616....2710413....5567530
..200..4059..34350..181336...710976...2269938...6233356...15250675...34054592
..351..9749.102896..644721..2908797..10462235..31868448...85473225..207289059
..616.23422.308419.2294193.11911516..48259083.163014678..479101189.1261310492
.1081.56268.924532.8166441.48807427.222798408.834763824.2688814689.7684922749

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A005251(n+5).

Row 1 is A002412(n+1).

Empirical for columns:
k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3)
k=2: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -a(n-4) +a(n-5)
k=3: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)
k=4: a(n) = 5*a(n-1) -10*a(n-2) +20*a(n-3) -15*a(n-4) +21*a(n-5) -7*a(n-6) +8*a(n-7) -a(n-8) +a(n-9)
k=5: a(n) = 6*a(n-1) -15*a(n-2) +35*a(n-3) -35*a(n-4) +56*a(n-5) -28*a(n-6) +36*a(n-7) -9*a(n-8) +10*a(n-9) -a(n-10) +a(n-11)
k=6: a(n) = 7*a(n-1) -21*a(n-2) +56*a(n-3) -70*a(n-4) +126*a(n-5) -84*a(n-6) +120*a(n-7) -45*a(n-8) +55*a(n-9) -11*a(n-10) +12*a(n-11) -a(n-12) +a(n-13)
k=7: a(n) = 8*a(n-1) -28*a(n-2) +84*a(n-3) -126*a(n-4) +252*a(n-5) -210*a(n-6) +330*a(n-7) -165*a(n-8) +220*a(n-9) -66*a(n-10) +78*a(n-11) -13*a(n-12) +14*a(n-13) -a(n-14) +a(n-15)
Empirical for rows:
n=1: a(k) = (2/3)*k^3 + (5/2)*k^2 + (17/6)*k + 1
n=2: a(k) = (1/3)*k^4 + (7/3)*k^3 + (14/3)*k^2 + (11/3)*k + 1
n=3: a(k) = (2/15)*k^5 + (11/6)*k^4 + (35/6)*k^3 + (23/3)*k^2 + (68/15)*k + 1
n=4: a(k) = (2/45)*k^6 + (19/15)*k^5 + (217/36)*k^4 + (71/6)*k^3 + (2057/180)*k^2 + (27/5)*k + 1
n=5: a(k) = (4/315)*k^7 + (7/9)*k^6 + (241/45)*k^5 + (1067/72)*k^4 + (3757/180)*k^3 + (1145/72)*k^2 + (2629/420)*k + 1
n=6: a(k) = (1/315)*k^8 + (134/315)*k^7 + (21/5)*k^6 + (571/36)*k^5 + (1841/60)*k^4 + (6047/180)*k^3 + (26603/1260)*k^2 + (299/42)*k + 1
n=7: a(k) = (2/2835)*k^9 + (131/630)*k^8 + (2803/945)*k^7 + (1349/90)*k^6 + (41449/1080)*k^5 + (20423/360)*k^4 + (1149293/22680)*k^3 + (22741/840)*k^2 + (2011/252)*k + 1

A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

Original entry on oeis.org

0, 1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, 103411, 124223, 148028, 175100, 205725, 240201, 278838, 321958, 369895, 422995, 481616, 546128, 616913, 694365, 778890

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Row sums of A103219.
From Bruno Berselli, Dec 10 2010: (Start)
a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.
The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)
a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - Vaclav Kotesovec, Dec 22 2011
Also, one-half the even-indexed terms of the partial sums of A045947. - J. M. Bergot, Apr 12 2018

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. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.

for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))

CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* Harvey P. Dale, Jan 23 2016 *)

a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+8*x+3*x^2)/(1-x)^5.
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013
a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - J. M. Bergot, Jun 10 2017
From Wesley Ivan Hurt, Nov 20 2021: (Start)
a(n) = Sum_{k=1..n} k * C(2*k,2).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)
From Peter Bala, Sep 03 2023: (Start)
a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).
Second subdiagonal of A039755. (End)

A103450 A figurate number triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 22, 22, 9, 1, 1, 11, 35, 50, 35, 11, 1, 1, 13, 51, 95, 95, 51, 13, 1, 1, 15, 70, 161, 210, 161, 70, 15, 1, 1, 17, 92, 252, 406, 406, 252, 92, 17, 1, 1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1, 1, 21, 145, 525, 1170, 1722, 1722, 1170, 525, 145, 21, 1

Offset: 0

Paul Barry, Feb 06 2005

Row coefficients are the absolute values of the coefficients of the characteristic polynomials of the n X n matrices A(n) with A(n){i,i} = 2, i>0, A(n){i,j} = 1, otherwise (starts with (0,0) position).

The triangle can be generated by the matrix multiplication A007318 * A114219s, where A114219s = 0; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,3,1; 0,-1,2,-3,4,1; ... = A097807 * A128229 is a signed variant of A114219. - Gary W. Adamson, Feb 20 2007

			From _Roger L. Bagula_, Oct 21 2008: (Start)
The triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  7,  12,   7,   1;
  1,  9,  22,  22,   9,   1;
  1, 11,  35,  50,  35,  11,   1;
  1, 13,  51,  95,  95,  51,  13,   1;
  1, 15,  70, 161, 210, 161,  70,  15,   1;
  1, 17,  92, 252, 406, 406, 252,  92,  17,  1;
  1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1; ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 9.
G. Chiaselotti, W. Keith, and P. A. Oliverio, Two Self-Dual Lattices of Signed Integer Partitions, Appl. Math. Inf. Sci. 8, No. 6, 3191-3199 (2014), via ResearchGate.

Row sums are A045623.
Columns include: A000326, A002412, A002418, A005408.
Cf. A000045, A208354.

A103450:= func< n,k | k eq 0 select 1 else Binomial(n, k)*(k*(n-k) + n)/n >;
[A103450(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 17 2021

(* First program *)
p[x_, n_]:= p[x, n]= If[n==0, 1, (-1+x)^(n-2)*(1 -(n+1)*x +x^2)];
T[n_, k_]:= T[n,k]= (-1)^(n+k)*SeriesCoefficient[p[x, n], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 21 2008 *)(* corrected by G. C. Greubel, Jun 17 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Binomial[n, k]*(n*(k+1) -k^2)/n];
Table[T[n, k], {n,0,16}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 17 2021 *)

def A103450(n, k): return 1 if (k==0) else binomial(n, k)*(k*(n-k) + n)/n
flatten([[A103450(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 17 2021

T(n, k) = binomial(n-1, k-1)*(k*(n-k) + n)/k with T(n, 0) = 1.
T(n, k) = T(n-1, k-1) + T(n-1, k) + binomial(n-2, k-1) with T(n, 0) = 1.
Column k is generated by (1+k*x)*x^k/(1-x)^(k+1).
Rows are coefficients of the polynomials P(0, x) = 1, P(n, x) = (1+x)^(n-2)*(1 +(n+1)*x + x^2) for n>0.
T(n,k) = Sum_{j=0..n} binomial(k, k-j)*binomial(n-k, j)*(j+1). - Paul Barry, Oct 28 2006
A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1, p(x, 1) = (-1 +x), p(x, 2) = (1 -3*x +x^2), p(x,n) = (-1 +x)^(n-2)*(1 - (n + 1)*x + x^2); T(n, k) = (-1)^(n+k)*coefficient of x^k of ( p(x,n) ). - Roger L. Bagula and Gary W. Adamson, Oct 21 2008
T(2*n+1, n) = A141222(n). - Emanuele Munarini, Jun 01 2012 [corrected by Werner Schulte, Nov 27 2021]
G.f.: is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). - Joerg Arndt, Aug 27 2013
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/5)*((-n+5)*Fibonacci(n+1) + (3*n- 2)*Fibonacci(n)) = A208354(n). - G. C. Greubel, Jun 17 2021
T(2*n, n) = A000984(n) * (n + 2) / 2 for n >= 0. - Werner Schulte, Nov 27 2021

A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

Original entry on oeis.org

0, 0, 2, 7, 17, 33, 57, 90, 134, 190, 260, 345, 447, 567, 707, 868, 1052, 1260, 1494, 1755, 2045, 2365, 2717, 3102, 3522, 3978, 4472, 5005, 5579, 6195, 6855, 7560, 8312, 9112, 9962, 10863, 11817, 12825, 13889, 15010, 16190, 17430, 18732, 20097, 21527

Offset: 0

R. K. Guy

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

First differences of A082289.

Cf. A000447, A002412, A051895.

[Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // Vincenzo Librandi, Aug 29 2011

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* Jean-François Alcover, Dec 12 2016 *)
CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 26 2021 *)

PARI
```
a(n)=(4*n^3+2*n^2-4*n)\16
```

G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - Michael Somos
a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - Bruno Berselli, Aug 29 2011
a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - J. M. Bergot, Apr 12 2018
E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - Stefano Spezia, Aug 22 2023

A132124 a(n) = n(n+1)(8*n + 1)/6.

Original entry on oeis.org

0, 3, 17, 50, 110, 205, 343, 532, 780, 1095, 1485, 1958, 2522, 3185, 3955, 4840, 5848, 6987, 8265, 9690, 11270, 13013, 14927, 17020, 19300, 21775, 24453, 27342, 30450, 33785, 37355, 41168, 45232, 49555, 54145, 59010, 64158, 69597, 75335, 81380, 87740, 94423

Offset: 0

Reinhard Zumkeller, Aug 12 2007

Convolution of the sequences (0,3,5,0,0,0,...) and (binomial(n+3, 3)), n >= 0. - Emeric Deutsch, Aug 30 2007

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A033994, A132112, A050409.
Cf. A014105, A016061; A002412, A006331; A007585, A002378.
Row sums of A069011.

seq((1/6)*n*(n+1)*(8*n+1),n=0..40); # Emeric Deutsch, Aug 30 2007

a[n_] := n*(n + 1)*(8*n + 1)/6; Array[a, 42, 0] (* Amiram Eldar, May 20 2023 *)

a(n) = A132121(n,2) for n > 1.
G.f.: x*(3+5*x)/(1-x)^4. - Emeric Deutsch, Aug 30 2007
From Bruno Berselli, Nov 25 2010: (Start)
a(n) = n*A014105(n) - A016061(n-1), since A016061(n-1) = Sum_{k=0..n-1} A014105(k) (n > 0).
Also a(n) = A002412(n) + A006331(n) = A007585(n) + A002378(n). (End)
Sum_{n>=1} 1/a(n) = 54 - 24*(sqrt(2)+1)*Pi/7 - 24*(sqrt(2)+8)*log(2)/7 + 48*sqrt(2)*log(2-sqrt(2))/7. - Amiram Eldar, May 20 2023
E.g.f.: exp(x)*x*(18 + 33*x + 8*x^2)/6. - Stefano Spezia, Feb 21 2024

A256645 25-gonal pyramidal numbers: a(n) = n(n+1)(23*n-20)/6.

Original entry on oeis.org

0, 1, 26, 98, 240, 475, 826, 1316, 1968, 2805, 3850, 5126, 6656, 8463, 10570, 13000, 15776, 18921, 22458, 26410, 30800, 35651, 40986, 46828, 53200, 60125, 67626, 75726, 84448, 93815, 103850, 114576, 126016, 138193, 151130, 164850, 179376, 194731, 210938, 228020

Offset: 0

Luciano Ancora, Apr 07 2015

If b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6 is n-th k-gonal pyramidal number, then b(n,k) = A000292(n) + (k-3)*A000292(n-1) (see Deza in References section, p. 96).
Also, b(n,k) = b(n,k-1) + A000292(n-1) (see Deza in References section, p. 95). Some examples:
  for k=4, A000330(n) = A000292(n) + A000292(n-1);
  for k=5, A002411(n) = A000330(n) + A000292(n-1);
  for k=6, A002412(n) = A002411(n) + A000292(n-1), etc.
This is the case k=25.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (23rd row of the table).

Colin Barker, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A255184.
Cf. similar sequences listed in A237616.
Cf. A000292, A000330, A002411, A002412, A256716.

k:=25; [n*(n+1)*((k-2)*n-(k-5))/6: n in [0..40]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1) (23 n - 20)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 26, 98}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

concat(0, Vec(x*(1 + 22*x)/(1 - x)^4 + O(x^100))) \\ Colin Barker, Apr 07 2015

G.f.: x*(1 + 22*x)/(1 - x)^4.
a(n) = A000292(n) + 22*A000292(n-1) = A256716(n) + A000292(n-1), see comments.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Apr 07 2015
E.g.f.: exp(x)*x*(6 + 72*x + 23*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

A162147 a(n) = n(n+1)(5*n + 4)/6.

Original entry on oeis.org

0, 3, 14, 38, 80, 145, 238, 364, 528, 735, 990, 1298, 1664, 2093, 2590, 3160, 3808, 4539, 5358, 6270, 7280, 8393, 9614, 10948, 12400, 13975, 15678, 17514, 19488, 21605, 23870, 26288, 28864, 31603, 34510, 37590, 40848, 44289, 47918, 51740, 55760

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Partial sums of A005475.
Suppose we extend the triangle in A215631 to a symmetric array by reflection about the main diagonal. The array is defined by m(i,j) = i^2 + i*j + j^2: 3, 7, 13, ...; 7, 12, 19, ...; 13, 19, 27, .... Then a(n) is the sum of the n-th antidiagonal. Examples: 3, 7 + 7, 13 + 12 + 13, 21 + 19 + 19 + 21, etc. - J. M. Bergot, Jun 25 2013
Binomial transform of [0,3,8,5,0,0,0,...]. - Alois P. Heinz, Mar 10 2015

			For n=4, a(4) = 0*(5+0) + 1*(5+1) + 2*(5+2) + 3*(5+3) + 4*(5+4) = 80. - _Bruno Berselli_, Mar 17 2016

Jinyuan Wang, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002412, A002413, A006331, A016061, A033994.

[n*(n+1)*(5*n+4)/6: n in [0..40]]; // G. C. Greubel, Apr 01 2021

A162147:= n-> n*(n+1)*(5*n+4)/6; seq(A162147(n), n=0..40); # G. C. Greubel, Apr 01 2021

Table[(n(n+1)(5n+4))/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,14,38},50] (* Harvey P. Dale, May 04 2013 *)

a(n)=n*(n+1)*(5*n+4)/6 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(5*n+4)/6 for n in (0..40)] # G. C. Greubel, Apr 01 2021

From R. J. Mathar, Jun 27 2009: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)
a(n) = A033994(n) + A000217(n).
G.f.: x*(3+2*x)/(1-x)^4. (End)
a(n) = A035005(n+1)/4. - Johannes W. Meijer, Feb 04 2010
a(n) = Sum_{i=0..n} i*(n + 1 + i). - Bruno Berselli, Mar 17 2016
E.g.f.: x*(18 + 24*x + 5*x^2)*exp(x)/6. - G. C. Greubel, Apr 01 2021

Definition rephrased by R. J. Mathar, Jun 27 2009

cp's OEIS Frontend

A093561 (4,1) Pascal triangle.

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A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

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A228461 Two-dimensional array read by antidiagonals: T(n,k) = number of arrays of maxima of three adjacent elements of some length n+2 0..k array.

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A200886 T(n,k) is the number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.

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A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.

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A103450 A figurate number triangle read by rows.

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A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

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A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

A132124 a(n) = n(n+1)(8*n + 1)/6.

A256645 25-gonal pyramidal numbers: a(n) = n(n+1)(23*n-20)/6.

A162147 a(n) = n(n+1)(5*n + 4)/6.