A060488 -id:A060488 - 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.

A095666 Pascal (1,4) triangle.

Original entry on oeis.org

4, 1, 4, 1, 5, 4, 1, 6, 9, 4, 1, 7, 15, 13, 4, 1, 8, 22, 28, 17, 4, 1, 9, 30, 50, 45, 21, 4, 1, 10, 39, 80, 95, 66, 25, 4, 1, 11, 49, 119, 175, 161, 91, 29, 4, 1, 12, 60, 168, 294, 336, 252, 120, 33, 4, 1, 13, 72, 228, 462, 630, 588, 372, 153, 37, 4, 1, 14, 85, 300, 690, 1092

Offset: 0

Wolfdieter Lang, Jun 11 2004

This is the fourth member, q=4, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1)), A029635 (q=2) (but with a(0,0)=2, not 1), A095660 (q=3), A096940 (q=5), A096956 (q=6).
This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column no. m of the type g(x)*(x*f(x))^m with f(0)=1. 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) = g(z)/(1 - x*z*f(z)). Here: g(x) = (4-3*x)/(1-x), f(x) = 1/(1-x), hence G(z,x) = (4-3*z)/(1-(1+x)*z).
The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k, k) = A022095(n-2), n >= 2, with n=1 value 4. [Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.]
T(2*n,n) = A029609(n) for n > 0, A029609 are the central terms of the Pascal (2,3) triangle A029600. - Reinhard Zumkeller, Apr 08 2012

			Triangle begins:
  [4];
  [1,4];
  [1,5,4];
  [1,6,9,4];
  [1,7,15,13,4];
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Wolfdieter Lang, First 10 rows.

Row sums: A020714(n-1), n >= 1, 4 if n=0.
Alternating row sums are [4, -3, followed by 0's].
Column sequences (without leading zeros) give for m=1..9, with n >= 0: A000027(n+4), A055999(n+1), A060488(n+3), A095667-71, A095819.

a095666 n k = a095666_tabl !! n !! k
a095666_row n = a095666_tabl !! n
a095666_tabl = [4] : iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,4]
-- Reinhard Zumkeller, Apr 08 2012

a(n,k):=(1+3*k/n)*binomial(n,k) # Mircea Merca, Apr 08 2012

A095666[n_, k_] := If[n == k,  4, (3*k/n + 1)*Binomial[n, k]];
Table[A095666[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Apr 14 2025 *)

Recursion: a(n, m) = 0 if m > n, a(0, 0) = 4; a(n, 0) = 1 if n>=1; a(n, m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (4-3*x)/(1-x)^(m+1), m >= 0.
a(n,k) = (1 + 3*k/n)*binomial(n,k). - Mircea Merca, Apr 08 2012

A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

Original entry on oeis.org

4, 60, 120, 13, 375, 3030, 9030, 5040, 28, 1392, 24552, 207900, 838320, 1345680, 362880, 50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600, 80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

All columns are polynomials of order binomial(k, 3). - Andrew Howroyd, Jan 30 2020

			Triangle begins:
  [4, 60, 120],
  [13, 375, 3030, 9030, 5040],
  [28, 1392, 24552, 207900, 838320, 1345680, 362880],
  [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
   ...
There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).

Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)

Row sums are A060491.
Columns k=4..6 are A060488, A060489, A060490.
Cf. A059443, A059530, A060052, A060092, A060487, A331571.

\\ gives g.f. of k-th column.
ColGf(k) = k!*polcoef(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^k) )*sum(j=0, k, 1/(1-y)^binomial(j, 3)*exp((-x^2/2)/(1-y)^j + O(x*x^k))*x^j/j!), k) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))); Mat([Col(p/y^3, -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ Andrew Howroyd, Jan 30 2020

E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y))*Sum_{k=0..inf}1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k!.

A026054 dot product (n,n-1,...2,1).(3,4,...,n,1,2).

Original entry on oeis.org

13, 28, 50, 80, 119, 168, 228, 300, 385, 484, 598, 728, 875, 1040, 1224, 1428, 1653, 1900, 2170, 2464, 2783, 3128, 3500, 3900, 4329, 4788, 5278, 5800, 6355, 6944, 7568, 8228, 8925, 9660, 10434, 11248, 12103, 13000, 13940, 14924, 15953, 17028, 18150, 19320, 20539, 21808, 23128, 24500, 25925

Offset: 3

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A023551.
Column 2 of triangle A094415.
Essentially the same as A060488. - Vladeta Jovovic, Jun 15 2006

List([0..60], n-> n*(n^2+9*n-10)/6); # G. C. Greubel, Oct 30 2019

[n*(n^2+9*n-10)/6: n in [3..60]]; // Vincenzo Librandi, Oct 17 2013

[n*(n^2+9*n-10)/6: n in [0..60]]; // G. C. Greubel, Oct 30 2019

seq(n*(n^2+9*n-10)/6, n=3..60); # G. C. Greubel, Oct 30 2019

Table[Range[n,1,-1].RotateLeft[Range[n],2],{n,3,60}] (* or *) LinearRecurrence[ {4,-6,4,-1},{13,28,50,80},60] (* Harvey P. Dale, Oct 14 2012 *)
Drop[CoefficientList[Series[x(13 -24x +16x^2 -4x^3)/(1-x)^4, {x, 0, 60}], x], 1] (* Vincenzo Librandi, Oct 17 2013 *)

vector(60, n, (n+2)*((n+2)^2+9*(n+2)-10)/6) \\ G. C. Greubel, Oct 30 2019

[n*(n^2+9*n-10)/6 for n in (0..60)] # G. C. Greubel, Oct 30 2019

a(n) = A023551(n+1) + 4.
From Colin Barker, Sep 17 2012: (Start)
a(n) = n*(n^2+9*n-10)/6.
G.f.: x^3*(13 - 24*x + 16*x^2 - 4*x^3)/(1-x)^4. (End)
E.g.f.: x^2*(-12 + (12+x)*exp(x))/6. - G. C. Greubel, Oct 30 2019

Closed-form formula corrected by Colin Barker, Sep 17 2012

A095667 Fifth column (m=4) of (1,4)-Pascal triangle A095666.

Original entry on oeis.org

4, 17, 45, 95, 175, 294, 462, 690, 990, 1375, 1859, 2457, 3185, 4060, 5100, 6324, 7752, 9405, 11305, 13475, 15939, 18722, 21850, 25350, 29250, 33579, 38367, 43645, 49445, 55800, 62744, 70312, 78540, 87465, 97125, 107559, 118807, 130910, 143910, 157850

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 4-subset of an n-set X then, for n>=7, a(n-7) is the number of 4-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007
In this sequence if we do a forward difference, then the 3rd forward difference when considered as a sequence will be an arithmetic progression with common difference 1. The same way the sequence formed by the 3rd forward difference of A047668 will be an arithmetic progression with common difference 8. [From Gopalakrishnan (gopala498(AT)yahoo.co.in), Jun 05 2010]
Row 4 of the convolution array A213550.  [Clark Kimberling, Jun 20 2012]

Partial sums of A060488.

A095667:=n->(n+16)*binomial(n+3,3)/4; seq(A095667(k), k=0..70); # Wesley Ivan Hurt, Sep 25 2013

s1=s2=s3=s4=0;lst={};Do[a=n+(n+2);s1+=a;s2+=s1;s3+=s2;s4+=s3;AppendTo[lst,s3/2],{n,3,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)

G.f.: (4-3*x)/(1-x)^5.
a(n) = 4*b(n)-3*b(n-1) = (n+16)*binomial(n+3, 3)/4, with b(n):=binomial(n+4, 4)= A000332(n+4, 4).
a(n) = sum_{k=1..n} ( sum_{i=1..k} i*(n-k+4) ). - Wesley Ivan Hurt, Sep 25 2013

A060091 Number of 4-block ordered bicoverings of an unlabeled n-set.

Original entry on oeis.org

0, 0, 0, 16, 63, 162, 341, 636, 1092, 1764, 2718, 4032, 5797, 8118, 11115, 14924, 19698, 25608, 32844, 41616, 52155, 64714, 79569, 97020, 117392, 141036, 168330, 199680, 235521, 276318, 322567, 374796, 433566, 499472, 573144, 655248, 746487

Offset: 0

Vladeta Jovovic, Feb 26 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Column k=4 of A060092.

Cf. A059945, A060090, A060488.

a(n) = if(n<1, 0, binomial(n + 5, 5) - 4*binomial(n + 2, 2) - 3*binomial(n + 1, 1) + 12*binomial(n, 0) - 6*binomial(n - 1, -1)) \\ Harry J. Smith, Jul 01 2009

a(n) = binomial(n + 5, 5) - 4*binomial(n + 2, 2) - 3*binomial(n + 1, 1) + 12*binomial(n, 0) - 6*binomial(n - 1, -1).
G.f.: - y^3*( - 24*y^2 - 16 + 33*y + 6*y^3)/( - 1 + y)^6.
E.g.f. for ordered k-block bicoverings of an unlabeled n-set is: exp( - x - x^2/2*y/(1 - y))*Sum_{k>=0} 1/(1 - y)^binomial(k, 2)*x^k/k!.
a(n) = (n+5)*(n-1)*(n-2)*(n^2+13*n+72)/120, n>0. - R. J. Mathar, Aug 15 2017

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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A095666 Pascal (1,4) triangle.

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A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

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A026054 dot product (n,n-1,...2,1).(3,4,...,n,1,2).

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A095667 Fifth column (m=4) of (1,4)-Pascal triangle A095666.

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A060091 Number of 4-block ordered bicoverings of an unlabeled n-set.

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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.