A071238 -id:A071238 - 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.

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 19, 14, 5, 44, 37, 22, 7, 85, 76, 55, 30, 9, 146, 135, 108, 73, 38, 11, 231, 218, 185, 140, 91, 46, 13, 344, 329, 290, 235, 172, 109, 54, 15, 489, 472, 427, 362, 285, 204, 127, 62, 17, 670, 651, 600, 525, 434, 335, 236, 145, 70, 19, 891, 870, 813

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A100157
Antidiagonal sums:  A071238
row 1,  (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900
row 2,  (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941
row 3,  (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...6....19...44....85....146
3...14...37...76....135...218
5...22...55...108...185...290
7...30...73...140...235...362
9...38...91...172...285...434

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := 2 n - 1;
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}]  (* A213752 *)
Table[t[n, n], {n, 1, 40}] (* A100157 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A071238 *)

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) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.

A071245 a(n) = n(n-1)(2*n^2 + 1)/6.

Original entry on oeis.org

0, 0, 3, 19, 66, 170, 365, 693, 1204, 1956, 3015, 4455, 6358, 8814, 11921, 15785, 20520, 26248, 33099, 41211, 50730, 61810, 74613, 89309, 106076, 125100, 146575, 170703, 197694, 227766, 261145, 298065, 338768, 383504, 432531, 486115, 544530, 608058

Offset: 0

N. J. A. Sloane, Jun 12 2002

The first differences are given in A277228. - J. M. Bergot, Sep 14 2016

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..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A071238, A071244, A277228 (first differences).

[n*(n-1)*(2*n^2+1)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Table[n (n - 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)
CoefficientList[Series[(-3 x^2 - 4 x^3 - x^4)/(-1 + x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Sep 14 2016 *)

a(n)=n*(n-1)*(2*n^2+1)/6; \\ Joerg Arndt, Sep 04 2013

def A071245(n): return binomial(n,2)*(2*n^2+1)//3
[A071245(n) for n in range(41)] # G. C. Greubel, Aug 07 2024

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4; a(0)=0, a(1)=0, a(2)=3, a(3)=19, a(4)=66. - Yosu Yurramendi, Sep 03 2013
G.f.: x^2*(3 + 4*x + x^2)/(1-x)^5. - Michael De Vlieger, Sep 14 2016
E.g.f.: (1/6)*x^2*(9 + 10*x + 2*x^2)*exp(x). - G. C. Greubel, Sep 23 2016

A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 2, 22, 13, 3, 50, 37, 19, 4, 95, 78, 52, 25, 5, 161, 140, 106, 67, 31, 6, 252, 227, 185, 134, 82, 37, 7, 372, 343, 293, 230, 162, 97, 43, 8, 525, 492, 434, 359, 275, 190, 112, 49, 9, 715, 678, 612, 525, 425, 320, 218, 127

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213837.
Antidiagonal sums: A071238.
Row 1, (1,5,9,13,...)**(1,2,3,4,...): A002412.
Row 2, (1,5,9,13,...)**(2,3,4,5,...): (4*k^3 + 15*k^2 - 7*k)/6.
Row 3, (1,5,9,13,...)**(3,4,5,6,...): (4*k^3 + 27*k^2 - 13*k)/6.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1...7....22...50....95
2...13...37...78....140
3...19...52...106...185
4...25...67...134...230
5...31...82...162...275
6...37...97...190...320

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

Cf. A212500.

b[n_]:=4n-3;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}] (* A213836 *)
Table[t[n,n],{n,1,40}] (* A213837 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A071238 *)

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) = x*(n + (2*n+1)*x - 3*(n-1)*x^2) and g(x) = (1-x)^4.

A071270 a(n) = n^2(2n^2 + 1)/3.

Original entry on oeis.org

0, 1, 12, 57, 176, 425, 876, 1617, 2752, 4401, 6700, 9801, 13872, 19097, 25676, 33825, 43776, 55777, 70092, 87001, 106800, 129801, 156332, 186737, 221376, 260625, 304876, 354537, 410032, 471801, 540300, 616001, 699392, 790977, 891276, 1000825, 1120176

Offset: 0

N. J. A. Sloane, Jun 12 2002

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..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A001105, A071238.

[n^2*(2*n^2+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

A071270:=n->(n^2)*(2*n^2+1)/3; seq(A071270(n), n=0..100); # Wesley Ivan Hurt, Nov 14 2013

Table[(n^2)(2n^2+1)/3, {n,0,100}] (* Wesley Ivan Hurt, Nov 14 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,57,176},50] (* Harvey P. Dale, Jan 09 2016 *)

a <- c(0, 1, 12, 57, 176)
for(n in (length(a)+1):30)
     a[n] <- 5*a[n-1]-10*a[n-2]+10*a[n-3]-5*a[n-4]+a[n-5]
a # Yosu Yurramendi, Sep 03 2013

def A071270(n): return binomial(2*n^2 + 1,2)/3
[A071270(n) for n in range(41)] # G. C. Greubel, Sep 13 2024

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=12, a(3)=57, a(4)=176. - Yosu Yurramendi, Sep 03 2013
a(n) = A000217(A001105(n))/ 3. - Michel Marcus, Mar 02 2018
From G. C. Greubel, Sep 13 2024: (Start)
G.f.: x*(1 + 7*x + 7*x^2 + x^3)/(1-x)^5.
E.g.f.: (1/3)*x*(3 + 15*x + 12*x^2 + 2*x^3)*exp(x). (End)

A277229 Convolution of the odd-indexed triangular numbers (A000384(n+1)) and the squares (A000290).

Original entry on oeis.org

0, 1, 10, 48, 158, 413, 924, 1848, 3396, 5841, 9526, 14872, 22386, 32669, 46424, 64464, 87720, 117249, 154242, 200032, 256102, 324093, 405812, 503240, 618540, 754065, 912366, 1096200, 1308538, 1552573, 1831728, 2149664, 2510288, 2917761, 3376506, 3891216

Offset: 0

Wolfdieter Lang, Oct 20 2016

This sequence was originally proposed in a comment on A071238 by J. M. Bergot as giving the first differences. Therefore, a(n) gives the partial sums of A071238.

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

Cf. A000217, A000290, A000384, A071238.

Table[n (n + 1) (n + 2) (4 n^2 + 3 n + 3)/60, {n, 0, 40}] (* Bruno Berselli, Oct 21 2016 *)

concat(0, Vec(x*((1+x)*(1+3*x))/(1-x)^6 + O(x^50))) \\ Colin Barker, Oct 21 2016

O.g.f.: x*(1 + x)*(1 + 3*x)/(1 - x)^6 = ((1 + 3*x)/(1 - x)^3)*(x*(1 + x)/(1 - x)^3).
a(n) = Sum_{k=0..n} A000384(n+1-k)*A000290(k).
a(n) = binomial(n+2, 3)*(4*n^2 + 3*n + 3)/10 = n*(n + 1)*(n + 2)*(4*n^2 + 3*n + 3)/60.
a(n) = Sum_{k=0..n} A071238(k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - Colin Barker, Oct 21 2016

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A071245 a(n) = n*(n-1)*(2*n^2 + 1)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A071270 a(n) = n^2*(2*n^2 + 1)/3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

R

SageMath

Formula

A277229 Convolution of the odd-indexed triangular numbers (A000384(n+1)) and the squares (A000290).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

A071245 a(n) = n(n-1)(2*n^2 + 1)/6.

A071270 a(n) = n^2(2n^2 + 1)/3.