A051662 -id:A051662 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 5, 2, 0, 0, 1, 4, 12, 18, 3, 0, 0, 1, 5, 22, 64, 66, 6, 0, 0, 1, 6, 35, 156, 363, 266, 12, 0, 0, 1, 7, 51, 310, 1193, 2214, 1111, 25, 0, 0, 1, 8, 70, 542, 2980, 9748, 14043, 4792, 52, 0, 0, 1, 9, 92, 868, 6273, 30526, 82916, 91857, 21124, 113, 0

Offset: 0

Alois P. Heinz, Feb 24 2015

From Vaclav Kotesovec, Feb 24 2015: (Start)
k    Limit n->infinity A(n,k)^(1/n)
  2.517540352632003890795354598463447277335981266803... = A246169
  5.249032491228170579164952216184309265343086337648... = A246312
  7.969494030514425004826375511986491746399264355846...
 10.688492754969652458452048798468242930479212456958...
 13.407087472537747579787047072702638639945914705837...
 16.125529360448558670505097146631763969697822205298...
 18.843901825822305757579605844910623225182677164912...
 21.562238702430237066018783115405680041128676137631...
 24.280555694806692616578932533497629224907619468796...
26.998860838916733933849490675388336975888308433826...
271.64425688361559470587959030374804709717287744789...
Conjecture: For big k the limit asymptotically approaches k*exp(1).
(End)

			A(3,2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2
Square array A(n,k) begins:
  0,  0,   0,    0,    0,     0,     0, ...
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  1,   5,   12,   22,    35,    51, ...
  0,  2,  18,   64,  156,   310,   542, ...
  0,  3,  66,  363, 1193,  2980,  6273, ...
  0,  6, 266, 2214, 9748, 30526, 77262, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.
Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.
Lower diagonal gives A255523.
Cf. A242249, A256068.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A204213 T(n,k) = Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than k.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 9, 9, 1, 5, 16, 32, 21, 1, 6, 25, 78, 120, 51, 1, 7, 36, 155, 404, 473, 127, 1, 8, 49, 271, 1025, 2208, 1925, 323, 1, 9, 64, 434, 2181, 7167, 12492, 8034, 835, 1, 10, 81, 652, 4116, 18583, 51945, 72589, 34188, 2188, 1, 11, 100, 933, 7120, 41363, 164255, 387000, 430569, 147787, 5798

Offset: 1

R. H. Hardin, Jan 12 2012

Table starts
...1....1.....1......1.......1.......1........1........1........1.........1
...2....3.....4......5.......6.......7........8........9.......10........11
...4....9....16.....25......36......49.......64.......81......100.......121
...9...32....78....155.....271.....434......652......933.....1285......1716
..21..120...404...1025....2181....4116.....7120....11529....17725.....26136
..51..473..2208...7167...18583...41363....82440...151125...259459....422565
.127.1925.12492..51945..164255..431445...991152..2057553..3945655...7098949
.323.8034.72589.387000.1493142.4629851.12262470.28832499.61766005.122779448

			Some solutions for n=5 k=3
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..3....0....1....2....2....2....1....3....0....3....1....2....2....3....2....1
..5....0....4....5....1....1....4....4....2....4....0....4....5....1....0....1
..5....1....2....4....3....0....1....1....3....1....1....2....5....3....2....2
..2....3....1....2....0....2....0....0....1....3....1....2....2....1....2....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

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

Column 1 is A001006; column 2 is A104184; column 3 is A204208.

Row 4 is A051662.

T[n_, k_] := T[n, k] = If[n == 0, 1, Sum[(Sum[Binomial[i, j]*(-1)^j* Binomial[-j*(2*k + 1) + i*(k + 1) - 1, i*k - j*(2*k + 1)], {j, 0, (i*k)/(2*k + 1)}])*T[n - i, k], {i, 1, n}]/n];
Table[T[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

T(n,k):=if n=0 then 1 else sum((sum(binomial(i,j)*(-1)^j*binomial(-j*(2*k+1)+i*(k+1)-1,i*k-j*(2*k+1)),j,0,(i*k)/(2*k+1)))*T(n-i,k),i,1,n)/n;  /* Vladimir Kruchinin, Apr 06 2017 */

{L(n,k)=polcoeff( ( (1-x^(2*k+1))/(1-x) +x*O(x^(k*n)) )^n, k*n)}
{T(n,k)=polcoeff(exp(sum(m=1, n, L(m,k)*x^m/m)+x*O(x^n)), n)}
for(n=1, 10,for(k=1,10, print1(T(n,k), ", "));print("")) \\ Paul D. Hanna, Aug 01 2013

Empirical for rows:
T(1,k) = 1
T(2,k) = k + 1
T(3,k) = k^2 + 2*k + 1
T(4,k) = (4/3)*k^3 + (7/2)*k^2 + (19/6)*k + 1
T(5,k) = (23/12)*k^4 + (37/6)*k^3 + (91/12)*k^2 + (13/3)*k + 1
T(6,k) = (44/15)*k^5 + (133/12)*k^4 + 17*k^3 + (161/12)*k^2 + (167/30)*k + 1
T(7,k) = (841/180)*k^6 + (101/5)*k^5 + (1325/36)*k^4 + (73/2)*k^3 + (946/45)*k^2 + (34/5)*k + 1
...
G.f. for column k: exp( Sum_{n>=1} L(n,k)*x^n/n ) - 1, where L(n,k) = central coefficient of (1+x+x^2+x^3+...+x^(2*k))^n. - Paul D. Hanna, Aug 01 2013
T(n,k):=Sum_{i=1..n}((Sum_{j=0..(i*k)/(2*k+1)}(binomial(i,j)*(-1)^j*binomial(-j*(2*k+1)+i*(k+1)-1,i*k-j*(2*k+1))))*T(n-i,k))/n, T(0,k)=1. - Vladimir Kruchinin, Apr 06 2017

A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

Original entry on oeis.org

0, 1, 12, 47, 120, 245, 436, 707, 1072, 1545, 2140, 2871, 3752, 4797, 6020, 7435, 9056, 10897, 12972, 15295, 17880, 20741, 23892, 27347, 31120, 35225, 39676, 44487, 49672, 55245, 61220, 67611, 74432, 81697, 89420, 97615, 106296, 115477, 125172

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - Jonathan Vos Post, May 27 2010

			a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - _Jonathan Vos Post_, May 27 2010

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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

Cf. A002378, A004466, A005915, A051662, A100145, A100188.

Cf. A214659, A214661.

[n*(n*(7*n-6)+2)/3: n in [0..50]]; // Vincenzo Librandi, May 12 2011

A051673:=n->n*(n*(7*n-6)+2)/3; seq(A051673(n), n=0..40); # Wesley Ivan Hurt, Feb 02 2014

Table[n^3+4Sum[i^2,{i,0,n-1}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,12,47},40] (* Harvey P. Dale, Jul 22 2011 *)

a(n)=n*(n*(7*n-6)+2)/3 \\ Charles R Greathouse IV, Oct 07 2015

[n*(7*n^2-6*n+2)/3 for n in range(51)] # G. C. Greubel, Mar 10 2024

a(n) = n*(n*(7*n-6) + 2)/3.
G.f.: x*(1+8*x+5*x^2)/(1-x)^4. - Bruno Berselli, May 12 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - Harvey P. Dale, Jul 22 2011
From Reinhard Zumkeller, Jul 25 2012: (Start)
a(n) = A214659(n) - A002378(n).
a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)
E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 10 2024

Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.

Original entry on oeis.org

1, 15, 62, 162, 335, 601, 980, 1492, 2157, 2995, 4026, 5270, 6747, 8477, 10480, 12776, 15385, 18327, 21622, 25290, 29351, 33825, 38732, 44092, 49925, 56251, 63090, 70462, 78387, 86885, 95976, 105680, 116017, 127007, 138670, 151026, 164095, 177897, 192452

Offset: 0

Bruno Berselli, Dec 11 2012

Sequence related to heptagonal pyramidal numbers (A002413) by a(n) = n*A002413(n) - (n-1)*A002413(n-1).
Other sequences of numbers of the form m*P(k,m)-(m-1)*P(k,m-1), where P(k,m) is the m-th k-gonal pyramidal number:
k=3, A002412(m) = m*A000292(m)-(m-1)*A000292(m-1);
k=4, A051662(m) = (m+1)*A000330(m+1)-m*A000330(m);
k=5, A213772(m) = m*A002411(m)-(m-1)*A002411(m-1);
k=6, A213837(m) = m*A002412(m)-(m-1)*A002412(m-1);
k=7, this sequence;
k=8, A130748(m) = m*A002414(m)-(m-1)*A002414(m-1).
Also, first bisection of A212983.
Binomial transform of (1, 14, 33, 20, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

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

Cf. A000292, A000330, A000566, A002411, A002412, A002413, A002414, A051662, A130748, A212983, A213772, A213837.

[(n+1)*(20*n^2+19*n+6)/6: n in [0..40]]; // Bruno Berselli, Jun 28 2016

/* By first comment: */  A002413:=func; [n*A002413(n)-(n-1)*A002413(n-1): n in [1..40]];

I:=[1,15,62,162]; [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..50]]; // Vincenzo Librandi, Aug 18 2013

Table[(n + 1) (20 n^2 + 19 n + 6)/6, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{1,15,62,162},40] (* Harvey P. Dale, Dec 23 2012 *)
CoefficientList[Series[(1 + 11 x + 8 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist((n+1)*(20*n^2+19*n+6)/6, n, 0, 20); /* Martin Ettl, Dec 12 2012 */

a(n)=(n+1)*(20*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+11*x+8*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3, a(0)=1, a(1)=15, a(2)=62, a(3)=162. - Harvey P. Dale, Dec 23 2012
a(n) = (n+1)*A000566(n+1) + Sum_{i=0..n} A000566(i). - Bruno Berselli, Dec 18 2013
E.g.f.: exp(x)*(6 + 84*x + 99*x^2 + 20*x^3)/6. - Elmo R. Oliveira, Aug 06 2025

A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

Original entry on oeis.org

1, 10, 36, 88, 175, 306, 490, 736, 1053, 1450, 1936, 2520, 3211, 4018, 4950, 6016, 7225, 8586, 10108, 11800, 13671, 15730, 17986, 20448, 23125, 26026, 29160, 32536, 36163, 40050, 44206, 48640, 53361, 58378, 63700, 69336, 75295, 81586, 88218, 95200, 102541

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999

Also as a(n) = (1/6)*(9*n^3 - 3*n^2), n>0: structured pentagonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of inequivalent tetrahedral edge colorings using at most n+1 colors so that no color appears only once. - David Nacin, Feb 22 2017

			        *     *
a(2) = * * + * * = 10.
       * *   * *

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000578, A051662, A100145, A100177.

Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.

[(3*n+2)*(n+1)^2/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

Table[((1+n)^2*(2+3n))/2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,10,36,88},40] (* Harvey P. Dale, Jun 26 2011 *)

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

a(n) = A000578(n+1) + (n+1)*A000217(n).
a(n) = (1/2)*(3*n+2)*(n+1)^2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=10, a(2)=36, a(3)=88. - Harvey P. Dale, Jun 26 2011
G.f.: (1+6*x+2*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n} (n+1)*(3*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=0} 1/a(n) = 9*log(3) - sqrt(3)*Pi - Pi^2/3 = 1.15624437161388... . - Vaclav Kotesovec, Oct 04 2016
E.g.f.: exp(x)*(2 + 18*x + 17*x^2 + 3*x^3)/2. - Elmo R. Oliveira, Aug 06 2025

A213751 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 2, 14, 9, 3, 30, 23, 13, 4, 55, 46, 32, 17, 5, 91, 80, 62, 41, 21, 6, 140, 127, 105, 78, 50, 25, 7, 204, 189, 163, 130, 94, 59, 29, 8, 285, 268, 238, 199, 155, 110, 68, 33, 9, 385, 366, 332, 287, 235, 180, 126, 77, 37, 10, 506, 485, 447, 396, 336, 271

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A051662
Antidiagonal sums:  A006325
row 1,  (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330
row 2,  (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986
row 3,  (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...5....14...30....55....91
2...9....23...46....80....127
3...13...32...62....105...163
4...17...41...78....130...199
5...21...50...94....155...235
6...25...59...110...180...271

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := n;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213751 *)
Table[t[n, n], {n, 1, 40}] (* A051662 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A006325 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = n + x - (n - 1)*x^2 and g(x) = (1 - x )^4.

A284551 Triangular array read by rows, demonstrating that the difference between a pentagonal number (left edge of triangle) and a square (right edge) is a triangular number.

Original entry on oeis.org

1, 5, 4, 12, 11, 9, 22, 21, 19, 16, 35, 34, 32, 29, 25, 51, 50, 48, 45, 41, 36, 70, 69, 67, 64, 60, 55, 49, 92, 91, 89, 86, 82, 77, 71, 64, 117, 116, 114, 111, 107, 102, 96, 89, 81, 145, 144, 142, 139, 135, 130, 124, 117, 109, 100, 176, 175, 173, 170, 166, 161, 155, 148, 140, 131, 121, 210, 209

Offset: 1

David Shane, Mar 29 2017

			Rows: {1}; {5,4}; {12,11,9}; ...
Triangle begins:
               1
            5     4
        12    11     9
     22    21    19    16
  35    34    32    29    25

Cf. A049777, A049780, which have a similar layout based on subtracting triangular numbers of increasing value from the leftmost element of the row.

A051662 gives row sums.

A284551 := proc(n,m)
    n*(3*n-1)-m*(m-1) ;
    %/2 ;
end proc:
seq(seq(A284551(n,m),m=1..n),n=1..15) ; # R. J. Mathar, Mar 30 2017

T[n_,m_]:= Floor[n(3n - 1) - m(m - 1)]/2; Table[T[n, k], {n, 12}, {k, n}] // Flatten (* Indranil Ghosh, Mar 30 2017 *)

T(n,m) = floor(n*(3*n - 1) - m*(m - 1))/2;
for(n=1, 12, for(k=1, n, print1(T(n,k),", ");); print();); \\ Indranil Ghosh, Mar 30 2017

def T(n, m): return (n*(3*n - 1) - m*(m - 1))//2
for n in range(1, 13):
    print([T(n,k) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 30 2017

P(m,n) = (m(3m-1) - n(n-1))/2. Alternatively, P(n) - T(n-1) = S(n) where P(n) is a pentagonal number, T(n-1) is a triangular number, and S(n) is a square number.

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A204213 T(n,k) = Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than k.

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A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

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A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.

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A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.