A045943 -id:A045943 - cp's OEIS frontend

A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 75, 52, 18, 4, 1, 541, 375, 130, 30, 5, 1, 4683, 3246, 1125, 260, 45, 6, 1, 47293, 32781, 11361, 2625, 455, 63, 7, 1, 545835, 378344, 131124, 30296, 5250, 728, 84, 8, 1, 7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 9, 1

Offset: 0

Mats Granvik, Jan 17 2009

Previous name: Matrix inverse of A154926.
A000670 appears in the first column. A052882 appears in the second column. A000027 and A045943 appear as diagonals. An alternative to calculating the matrix inverse of A154926 is to move the term in the lower right corner to a position in the same column and calculate the determinant instead, which yields the same answer.
Matrix inverse of (2*I - P), where P is Pascal's triangle and I the identity matrix. See A162312 for the matrix inverse of (2*P - I) and some general remarks about arrays of the form M(a) := (I - a*P)^-1 and their connection with weighted sums of powers of integers. The present array equals (1/2)*M(1/2). - Peter Bala, Jul 01 2009
From Mats Granvik, Aug 11 2009: (Start)
The values in this triangle can be seen as permanents of the Pascal triangle analogous to the method in the Redheffer matrix. The elements satisfy (T(n,k)/T(n,k-1))*k = (T(n-1,k)/T(n,k))*n which converges to log(2) as n->oo and k->0. More generally to calculate log(x) multiply the negative values in A154926 by 1/(x-1) and calculate the matrix inverse. Then (T(n,k)/T(n,k-1))*k and (T(n-1,k)/T(n,k))*n in the resulting triangle converge to log(x).
This method for calculating log(x) converges faster than the Taylor series when x is greater than 5 or so. See chapter on Taylor series in Spiegel for comparison. (End)
Exponential Riordan array [1/(2-exp(x)),x]. - Paul Barry, Apr 06 2011
T(n,k) is the number of ordered set partitions of {1,2,...,n} such that the first block contains k elements.  For k=0 the first block contains arbitrarily many elements. - Geoffrey Critzer, Jul 22 2013
A natural (signed) refinement of these polynomials is given by the Appell sequence e.g.f. e^(xt)/ f(t) = exp[tP.(x)] with the formal Taylor series f(x) = 1 + x[1] x + x[2] x^2/2! + ... and with raising operator R = x - d[log(f(D)]/dD (cf. A263634). - Tom Copeland, Nov 06 2015

			From _Peter Bala_, Jul 01 2009: (Start)
Triangle T(n, k) begins:
n\k|     0     1     2     3     4     5     6
==============================================
0  |     1
1  |     1     1
2  |     3     2     1
3  |    13     9     3     1
4  |    75    52    18     4     1
5  |   541   375   130    30     5     1
6  |  4683  3246  1125   260    45     6     1
...
(End)
From _Mats Granvik_, Aug 11 2009: (Start)
Row 4 equals 75,52,18,4,1 because permanents of:
  1,0,0,0,1  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0
  1,1,0,0,0  1,1,0,0,1  1,1,0,0,0  1,1,0,0,0  1,1,0,0,0
  1,2,1,0,0  1,2,1,0,0  1,2,1,0,1  1,2,1,0,0  1,2,1,0,0
  1,3,3,1,0  1,3,3,1,0  1,3,3,1,0  1,3,3,1,1  1,3,3,1,0
  1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,1
are:
     75         52         18          4          1
(End)

Murray R. Spiegel, Mathematical handbook, Schaum's Outlines, p. 111.

Alois P. Heinz, Rows n = 0..140, flattened
R. B. Nelsen, Problem E3062, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.
R. B. Nelsen and H. Schmidt, Jr., Chains in Power Sets, Mathematics Magazine, Vol. 64, No. 1 (Feb., 1991), 23-31.

Cf. A000629 (row sums), A000670, A007047, A052822 (column 1), A052841 (alt. row sums), A080253, A162312, A162313.

Cf. A263634, A099880 (T(2n,n)).

A154921_row := proc(n) local i,p; p := proc(n,x) option remember; local k;
if n = 0 then 1 else add(p(k,0)*binomial(n,k)*(1+x^(n-k)),k=0..n-1) fi end:
seq(coeff(p(n,x),x,i),i=0..n) end: for n from 0 to 5 do A154921_row(n) od;
# Peter Luschny, Jul 15 2012
T := (n,k) -> binomial(n,k)*add(combinat:-eulerian1(n-k,j)*2^j, j=0..n-k):
seq(print(seq(T(n,k), k=0..n)),n=0..6); # Peter Luschny, Feb 07 2015
# third Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
T:= (n, k)-> n!/k! *b(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 03 2019
# fourth Maple program:
p := proc(n, m) option remember; if n = 0 then 1 else
    (m + x)*p(n - 1, m) + (m + 1)*p(n - 1, m + 1) fi end:
row := n -> local k; seq(coeff(p(n, 0), x, k), k = 0..n):
for n from 0 to 6 do row(n) od;  # Peter Luschny, Jun 23 2023

nn = 8; a = Exp[x] - 1;
Map[Select[#, # > 0 &] &,
  Transpose[
   Table[Range[0, nn]! CoefficientList[
Series[x^n/n!/(1 - a), {x, 0, nn}], x], {n, 0, nn}]]] // Grid (* Geoffrey Critzer, Jul 22 2013 *)
E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
T[n_, k_] := Binomial[n, k] Sum[E1[n - k, j] 2^j, {j, 0, n - k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 30 2018, after Peter Luschny *)

@CachedFunction
def Poly(n, x):
    return 1 if n == 0 else add(Poly(k,0)*binomial(n,k)*(x^(n-k)+1) for k in range(n))
R = PolynomialRing(ZZ, 'x')
for n in (0..6): print(R(Poly(n,x)).list()) # Peter Luschny, Jul 15 2012

From Peter Bala, Jul 01 2009: (Start)
TABLE ENTRIES
  (1) T(n,k) = binomial(n,k)*A000670(n-k).
GENERATING FUNCTION
  (2) exp(x*t)/(2-exp(t)) = 1 + (1+x)*t + (3+2*x+x^2)*t^2/2! + ....
PROPERTIES OF THE ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. They appear in the study of the poset of power sets [Nelsen and Schmidt].
The first few values are R_0(x) = 1, R_1(x) = 1+x, R_2(x) = 3+2*x+x^2 and R_3(x) = 13+9*x+3*x^2+x^3.
The row polynomials may be recursively computed by means of
  (3) R_n(x) = x^n + Sum_{k = 0..n-1} binomial(n,k)*R_k(x).
Explicit formulas include
  (4) R_n(x) = (1/2)*Sum_{k >= 0} (1/2)^k*(x+k)^n,
  (5) R_n(x) = Sum_{j = 0..n} Sum_{k = 0..j} (-1)^(j-k)*binomial(j,k) *(x+k)^n,
and
  (6) R_n(x) = Sum_{j = 0..n} Sum_{k = j..n} k!*Stirling2(n,k) *binomial(x,k-j).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
  (7) 2*R_m(x) - R_m(x+1) = x^m,
which easily leads to the evaluation of the weighted sums of powers of integers
  (8) Sum_{k = 1..n-1} (1/2)^k*k^m = 2*R_m(0) - (1/2)^(n-1)*R_m(n).
For example, m = 2 gives
  (9) Sum_{k = 1..n-1} (1/2)^k*k^2 = 6 - (1/2)^(n-1)*(n^2+2*n+3).
More generally we have
  (10) Sum_{k=0..n-1} (1/2)^k*(x+k)^m = 2*R_m(x) - (1/2)^(n-1)*R_m(x+n).
RELATIONS WITH OTHER SEQUENCES
Sequences in the database given by particular values of the row polynomials are
  (11) A000670(n) = R_n(0)
  (12) A052841(n) = R_n(-1)
  (13) A000629(n) = R_n(1)
  (14) A007047(n) = R_n(2)
  (15) A080253(n) = 2^n*R_n(1/2).
This last result is the particular case (x = 0) of the result that the polynomials 2^n*R_n(1/2+x/2) are the row generating polynomials for A162313.
The above formulas should be compared with those for A162312. (End)
From Peter Luschny, Jul 15 2012: (Start)
  (16) A151919(n) = R_n(1/3)*3^n*(-1)^n
  (17) A052882(n) = [x^1] R_n(x)
  (18) A045943(n) = [x^(n-1)] R_n+1(x)
  (19) A099880(n) = [x^n] R_2n(x). (End)
The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1} binomial(n,k)*p{k}(0)*(1+x^(n-k)). - Peter Luschny, Jul 15 2012

New name by Peter Luschny, Feb 07 2015

A126804 a(n) = (2n)! / (n-1)!.

Original entry on oeis.org

2, 24, 360, 6720, 151200, 3991680, 121080960, 4151347200, 158789030400, 6704425728000, 309744468633600, 15543540607795200, 841941782922240000, 48962152914554880000, 3042648073975910400000, 201220459292273541120000, 14110584707870682071040000

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007

Old name was "Multiplying n X n integers above n".
a(n) = 2*A001814(n). - Zerinvary Lajos, May 03 2007
A179214(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
Product of the numbers from n to 2n. - Wesley Ivan Hurt, Dec 14 2015

			a(5) = 151200 because five digits above 5: (6, 7, 8, 9, 10), multiplied by five equals 5*(6*7*8*9*10) = 151200.

Robert Israel, Table of n, a(n) for n = 1..360
Elliot J. Carr and Matthew J. Simpson, Accurate and efficient calculation of finite response times for groundwater flow, arXiv:1707.06331 [physics.flu-dyn], 2017, p. 11.

Cf. A045943, A073838. - Reinhard Zumkeller, Jul 05 2010

Cf. A001814, A179214.

[Factorial(2*n)/Factorial(n-1) : n in [1..20]]; // Wesley Ivan Hurt, Dec 14 2015

a:=n->sum((count(Permutation(2*n+2),size=n+1)),j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(mul((n+k), k=0..n), n=1..16); # Zerinvary Lajos, Sep 21 2007
with(combstruct):with(combinat) :bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,labeled],size=n)*(n-1), n=2..17); # Zerinvary Lajos, Dec 05 2007

Table[Pochhammer[n, n + 1], {n, 17}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[(2 n)!/(n - 1)!, {n, 20}] (* Wesley Ivan Hurt, Dec 14 2015 *)

a(n) = prod(k=n, 2*n, k); \\ Michel Marcus, Dec 15 2015

x='x+O('x^99); Vec(serlaplace(2*x/(1-4*x)^(3/2))) \\ Altug Alkan, Mar 11 2018

a(n) = (2n)! / (n-1)!.
a(n) = Product_{i=n..2n} i. - Wesley Ivan Hurt, Dec 14 2015
From Robert Israel, Dec 15 2015: (Start)
a(n) = (2*n*(2*n-1)/(n-1))*a(n-1).
E.g.f.: 2*x/(1-4*x)^(3/2). (End)
a(n) = Pochhammer(n,n+1). - Pedro Caceres, Mar 10 2018

New name from Wesley Ivan Hurt, Dec 15 2015

A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

Original entry on oeis.org

0, 12, 27, 45, 66, 90, 117, 147, 180, 216, 255, 297, 342, 390, 441, 495, 552, 612, 675, 741, 810, 882, 957, 1035, 1116, 1200, 1287, 1377, 1470, 1566, 1665, 1767, 1872, 1980, 2091, 2205, 2322, 2442, 2565, 2691, 2820, 2952, 3087, 3225, 3366, 3510, 3657, 3807, 3960

Offset: 0

N. J. A. Sloane, May 15 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Leo Tavares, Illustration: Star Triangles
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n + 3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Cf. A003154, A060544.

s=0;lst={};Do[AppendTo[lst,s+=n],{n,12,6!,3}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 05 2010 *)
LinearRecurrence[{3,-3,1}, {0,12,27}, 50] (* or *) With[{nn = 50}, CoefficientList[Series[(3/2)*(8*x + x^2)*Exp[x], {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 26 2017 *)

x='x+O('x^50); concat([0], Vec(serlaplace((3/2)*(8*x + x^2)*exp(x)))) \\ G. C. Greubel, May 26 2017

a(n)=(3*n^2+21*n)/2 \\ Charles R Greathouse IV, Jun 16 2017

a(n) = a(n-1) + 3*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(4 - 3*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
From G. C. Greubel, May 26 2017: (Start)
a(n) = 3*n*(n+7)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*(8*x + x^2)*exp(x). (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 121/490.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/21 - 319/4410. (End)
a(n) = A003154(n+1) - A060544(n). - Leo Tavares, Mar 26 2022

A226492 a(n) = n(11n-5)/2.

Original entry on oeis.org

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523

Offset: 0

Bruno Berselli, Jun 11 2013

Sequences of numbers of the form n*(n*k - k + 6)/2:
. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
. k=11: a(n);
. k=12: A094159;
. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
. k=15: A152773;
. k=16: A139272;
. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
. k=18: A152751;
. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
. k=21: A152759;
. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
. k=24: A152767;
. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
. k=27: A153783;
. k=28: A195021;
. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
. k=30: A153448;
. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
. k=33: A153875.
Also:
a(n) - n         = A180223(n);
a(n) + n         = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) - 2*n       = A051865(n);
a(n) + 2*n       = A022268(n);
a(n) - 3*n       = A152740(n-1);
a(n) + 3*n       = A022269(n);
a(n) - 4*n       = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
a(n) + 4*n       = A254963(n);
a(n) - n*(n-1)/2 = A147874(n+1);
a(n) + n*(n-1)/2 = A094159(n) (case k=12);
a(n) - n*(n-1)   = A062741(n) (see above, this is the case k=9);
a(n) + n*(n-1)   = n*(13*n-7)/2 (case k=13);
a(n) - n*(n+1)/2 = A135706(n);
a(n) + n*(n+1)/2 = A033579(n);
a(n) - n*(n+1)   = A051682(n);
a(n) + n*(n+1)   = A186030(n);
a(n) - n^2       = A062708(n);
a(n) + n^2       = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. sequences in Comments lines.
First differences are in A017425.
Cf. A033584, A180223.

Magma
```
[n*(11*n-5)/2: n in [0..50]];
```

I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013

Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)

a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(3+8*x)/(1-x)^3.
a(n) + a(-n) = A033584(n).
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(6 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A180223(n). (End)

A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

Original entry on oeis.org

0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A033994, A129371, A132112, A132121, A132124.
Cf. A225144. - Bruno Berselli, Jun 06 2013
Cf. A045943: Sum_{k = n..2*n} k.
Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.

List([0..40], n-> n*(n+1)*(14*n+1)/6); # G. C. Greubel, Oct 30 2019

[&+[k^2: k in [n..2*n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011

I:=[0, 5, 29, 86]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

seq(add((n+k)^2,k=0..n),n=0..40); # Zerinvary Lajos, Dec 01 2006

LinearRecurrence[{4,-6,4,-1},{0,5,29,86},40] (* Vincenzo Librandi, Jun 22 2012 *)
Table[(n(n+1)(14n+1))/6,{n,0,40}] (* Harvey P. Dale, Mar 08 2020 *)

PARI
```
a(n)=sum(k=n,n+n,k^2)
```

vector(40, n, n*(n-1)*(14*n-13)/6) \\ G. C. Greubel, Oct 30 2019

[n*(n+1)*(14*n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = n*(n+1)*(14*n+1)/6.
a(n) = A132121(n,4) for n>3. - Reinhard Zumkeller, Aug 12 2007
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5+9*x)/(1-x)^4.
a(n) = A129371(2*n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012
E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

A140673 a(n) = 3n(n + 5)/2.

Original entry on oeis.org

0, 9, 21, 36, 54, 75, 99, 126, 156, 189, 225, 264, 306, 351, 399, 450, 504, 561, 621, 684, 750, 819, 891, 966, 1044, 1125, 1209, 1296, 1386, 1479, 1575, 1674, 1776, 1881, 1989, 2100, 2214, 2331, 2451, 2574, 2700, 2829, 2961, 3096

Offset: 0

Omar E. Pol, May 22 2008

a(n) equals the number of vertices of the A256666(n)-th graph (see Illustration of initial terms in A256666 Links). - Ivan N. Ianakiev, Apr 20 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A055998.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Table[Sum[i + n - 3, {i, 6, n}], {n, 5, 52}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[3 n (n + 5)/2, {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3,-3,1},{0,9,21},50] (* Harvey P. Dale, Jul 20 2023 *)

concat(0, Vec(3*x*(3 - 2*x)/(1 - x)^3 + O(x^100))) \\ Michel Marcus, Apr 20 2015

a(n) = 3*n*(n+5)/2; \\ Altug Alkan, Sep 05 2018

a(n) = A055998(n)*3 = (3*n^2 + 15*n)/2 = n*(3*n + 15)/2.
a(n) = 3*n + a(n-1) + 6 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(3 - 2*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 18*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 137/450.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 47/450. (End)

A140674 a(n) = n(3n + 17)/2.

Original entry on oeis.org

0, 10, 23, 39, 58, 80, 105, 133, 164, 198, 235, 275, 318, 364, 413, 465, 520, 578, 639, 703, 770, 840, 913, 989, 1068, 1150, 1235, 1323, 1414, 1508, 1605, 1705, 1808, 1914, 2023, 2135, 2250, 2368, 2489, 2613, 2740, 2870, 3003, 3139

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

[n*(3*n + 17)/2: n in [0..70]]; // Wesley Ivan Hurt, Apr 21 2021

Table[(3*n^2 + 17*n)/2, {n, 0, 50}] (* G. C. Greubel, Jul 17 2017 *)

a(n)=n*(3*n+17)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (3*n^2 + 17*n)/2.
a(n) = 7*n + 3*A000217(n). - Reinhard Zumkeller, May 28 2008
a(n) = 3*n + a(n-1) + 7 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: x*(10 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 20*x)*exp(x). - G. C. Greubel, Jul 17 2017

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Original entry on oeis.org

0, 3, 27, 72, 138, 225, 333, 462, 612, 783, 975, 1188, 1422, 1677, 1953, 2250, 2568, 2907, 3267, 3648, 4050, 4473, 4917, 5382, 5868, 6375, 6903, 7452, 8022, 8613, 9225, 9858, 10512, 11187, 11883, 12600, 13338, 14097, 14877, 15678, 16500, 17343, 18207, 19092, 19998

Offset: 0

Omar E. Pol, Dec 14 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001106, A139268, A226491.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=21: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,21}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
CoefficientList[Series[3 x (1 + 6 x) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{3,-3,1},{0,3,27},40] (* Harvey P. Dale, May 26 2015 *)

a(n)=3*n*(7*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (21*n^2 - 15*n)/2 = 3*A001106(n).
a(n) = a(n-1) + 21*n - 18 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+6*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = n + A226491(n). - Bruno Berselli, Jun 11 2013
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(2 + 7*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

Original entry on oeis.org

0, 3, 30, 81, 156, 255, 378, 525, 696, 891, 1110, 1353, 1620, 1911, 2226, 2565, 2928, 3315, 3726, 4161, 4620, 5103, 5610, 6141, 6696, 7275, 7878, 8505, 9156, 9831, 10530, 11253, 12000, 12771, 13566, 14385, 15228, 16095, 16986, 17901, 18840, 19803, 20790, 21801

Offset: 0

Omar E. Pol, Dec 15 2008

3*A172078(n) = n*a(n) - Sum_{k=0..n-1} a(k). - Bruno Berselli, Dec 12 2010

			For n=8, a(8) = (1*3 + 5*7 + 9*11 +..+ 29*31) - (2*4 + 6*8 + 10*12 +..+ 26*28) = 696 (see Problem 1052 in References). - _Bruno Berselli_, Dec 12 2010

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Jan. 1910 p. 47 (Problem 1052).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A001539, A002939, A139271, A153794, A172078.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=24: see Comments lines of A226492.

Table[3n(4n-3),{n,0,40}] (* Harvey P. Dale, May 26 2012 *)

a(n)=3*n*(4*n-3) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 12*n^2 - 9*n = 3*A001107(n).
a(n) = a(n-1) + 24*n - 21, n > 0. - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k=0..n-1} A001539(k) - Sum_{k=0..n-1} 4*A002939(k) if n > 0 (see References, Problem 1052). - Bruno Berselli, Dec 08 2010 - Jan 21 2011
G.f.: -3*x*(1+7*x)/(x-1)^3.
a(0)=0, a(1)=3, a(2)=30, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 26 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 4*x).
a(n) = A153794(n) - n. (End)

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

Original entry on oeis.org

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

cp's OEIS Frontend

A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.

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A126804 a(n) = (2n)! / (n-1)!.

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A151542 Generalized pentagonal numbers: a(n) = 12*n + 3*n*(n-1)/2.

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A226492 a(n) = n*(11*n-5)/2.

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A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

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A140673 a(n) = 3*n*(n + 5)/2.

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A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

A226492 a(n) = n(11n-5)/2.

A140673 a(n) = 3n(n + 5)/2.

A140674 a(n) = n(3n + 17)/2.

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).