A000579 -id:A000579 - cp's OEIS frontend

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

1, 16, 100, 400, 1225, 3136, 7056, 14400, 27225, 48400, 81796, 132496, 207025, 313600, 462400, 665856, 938961, 1299600, 1768900, 2371600, 3136441, 4096576, 5290000, 6760000, 8555625, 10732176, 13351716, 16483600, 20205025, 24601600, 29767936, 35808256

Offset: 0

N. J. A. Sloane

Total area of all square and rectangular regions from an n+1 X n+1 grid. E.g., n = 2, there are 9 individual squares, 4 2 X 2's and 1 3 X 3, total area 9 + 16 + 9 = 34. The rectangular regions include 6 2 X 1's, 6 1 X 2's, 3 3 X 1's, 3 1 X 3's, 2 3 X 2's and 2 2 X 3's, total area 12 + 12 + 9 + 9 + 12 + 12 = 66, hence a(2) = 34 + 66 = 100. - Jon Perry, Jul 29 2003 [Index/grid size adjusted by Rick L. Shepherd, Jun 27 2017]
Number of 3 X 3 submatrices of an n+3 X n+3 matrix. - Rick L. Shepherd, Jun 27 2017
The inverse binomial transform gives row n=2 of A087107. - R. J. Mathar, Aug 31 2022

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A000292, A006542, A033455, A108674 (first diffs.), A086020 (partial sums).
Third column of triangle A008459.
Cf. A000579, A001303, A040977.

A001249 := proc(n) binomial(n+3,n)^2 end proc: seq(A001249(n),n=0..10) ; # Zerinvary Lajos, May 17 2006

Table[Binomial[n + 3, 3]^2, {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

a(n)=binomial(n+3,3)^2 \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Aug 19 2008: (Start)
a(n) = (A000292(n+1))^2.
O.g.f.: (1+x)(x^2+8x+1)/(1-x)^7. (End)
a(n) = C(n+4, 3)*C(n+4, 4)/(n+4) + A001303(n) = C(n+4, 3)*C(n+3, 3)/4 + A001303(n) = C(n+4, 6) + 3*C(n+5, 6) + C(n+6,6) + A001303(n). - Gary Detlefs, Aug 07 2013
-n^2*a(n) + (n+3)^2*a(n-1) = 0. - R. J. Mathar, Aug 15 2013
a(n) = 9*A040977(n-1) + A000579(n+6) + A000579(n+3). - R. J. Mathar, Aug 15 2013
a(n) = (n+3)*C(n+2, 2)*C(n+3, 3)/3. - Gary Detlefs, Jan 06 2014
a(n) = A000290(n+1)*A000290(n+2)*A000290(n+3)/36. - Bruno Berselli, Nov 12 2014
G.f. 2F1(4,4;1;x). - R. J. Mathar, Aug 09 2015
E.g.f.: exp(x)*(1 + 15*x + 69*x^2/2! + 147*x^3/3! + 162*x^4/4! +  90*x^5/5! +  20*x^6/6!). Computed from the o.g.f with the formulas (23) - (25) of the W. Lang link given in A060187. - Wolfdieter Lang, Jul 27 2017
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 9*Pi^2 - 351/4.
Sum_{n>=0} (-1)^n/a(n) = 63/4 - 3*Pi^2/2. (End)
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Aug 29 2022
a(n) = a(n-1)+A000217(n+1)*A000330(n+1). - J. M. Bergot, Aug 29 2022
a(n) = A002415(n+2)^2 - 20*A006857(n-1). - Yasser Arath Chavez Reyes, Nov 08 2024

A035038 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 8, 37, 130, 386, 1024, 2510, 5812, 12911, 27824, 58651, 121670, 249528, 507624, 1026876, 2069256, 4158861, 8344056, 16721761, 33486026, 67025182, 134116144, 268313018, 536724316, 1073567387, 2147277280, 4294724471, 8589650318, 17179537972

Offset: 0

N. J. A. Sloane

Starting with "1", equals the eigensequence of a triangle with A000579 = binomial(n,6) = (1, 7, 28, 84, 210, ...) as the left column and the rest 1's. - Gary W. Adamson, Jul 24 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.

Cf. A000079, A000225, A000295, A000579, A002663, A002664, A007318.

Cf. A035039, A035040, A035041, A035042.

a035038 n = a035038_list !! n
a035038_list = map (sum . drop 6) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

[n le 5 select 0 else (&+[Binomial(n,j): j in [6..n]]): n in [0..50]]; // G. C. Greubel, Mar 20 2023

a:= n-> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [8,-27,50,-55, 36,-13,2][i] else 0 fi)^(n))[1,7]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 05 2008

Table[Sum[Binomial[n, k+6], {k,0,n}], {n,0,30}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[2^n-Total[Binomial[n,Range[0,5]]],{n,0,40}] (* Harvey P. Dale, Oct 24 2017 *)

[sum(binomial(n,j) for j in range(6,n+1)) for n in range(51)] # G. C. Greubel, Mar 20 2023

From Paul Barry, Aug 23 2004: (Start)
G.f.: x^6/((1-2*x)*(1-x)^6).
a(n) = Sum_{k=0..n} C(n, k+6) = Sum_{k=6..n} C(n, k).
a(n) = 2*a(n-1) + C(n-1, 5).  (End)

A085439 a(n) = Sum_{i=1..n} binomial(i+1,2)^4.

Original entry on oeis.org

1, 82, 1378, 11378, 62003, 256484, 871140, 2550756, 6651381, 15802006, 34776742, 71791798, 140366759, 261917384, 469277384, 811379400, 1359360681, 2214396762, 3517606762, 5462416762, 8309813083, 12406965164, 18209748140, 26309748140, 37466388765, 52644875166

Offset: 1

André F. Labossière, Jul 03 2003

			a(15) = (2520*(15^9) +22680*(15^8) +79920*(15^7) +136080*(15^6) +107352*(15^5) +22680*(15^4) -10080*(15^3) +1728*15)/9! = 469277384.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Column k=4 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

[(2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/Factorial(9): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[(2520*(n^9) + 22680*(n^8) + 79920*(n^7) + 136080*(n^6) + 107352*(n^5) + 22680*(n^4) - 10080*(n^3) + 1728*n)/9!, {n, 1, 50}] (* G. C. Greubel, Nov 22 2017 *)

Vec(x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1)/(x-1)^10 + O(x^100)) \\ Colin Barker, May 02 2014

a(n) = sum(i=1, n, binomial(i+1, 2)^4); \\ Michel Marcus, Nov 22 2017

a(n) = (2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/9!.

G.f.: x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1) / (x-1)^10. - Colin Barker, May 02 2014

More terms from Colin Barker, May 02 2014

Typo in example fixed by Colin Barker, May 02 2014

A085440 a(n) = Sum_{i=1..n} binomial(i+1,2)^5.

Original entry on oeis.org

1, 244, 8020, 108020, 867395, 4951496, 22161864, 82628040, 267156165, 770440540, 2022773116, 4909947484, 11150268935, 23913084560, 48796284560, 95322158736, 179163294729, 325374464580, 572984364580, 981394464580, 1639143014731, 2675722491224, 4277290592600

Offset: 1

André F. Labossière, Jun 30 2003

Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Column k=5 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

[(113400*n^11 +1247400*n^10 +5544000*n^9 +12474000*n^8 +14196600*n^7 +6237000*n^6 -831600*n^5 +1108800*n^3 -172800*n )/Factorial(11): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[(113400*n^11 +1247400*n^10 +5544000*n^9 +12474000*n^8 +14196600*n^7 +6237000*n^6 -831600*n^5 +1108800*n^3 -172800*n)/11!, {n,1,50}] (* G. C. Greubel, Nov 22 2017 *)

for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^5), ", ")) \\ G. C. Greubel, Nov 22 2017

a(n) = (113400*n^11 +1247400*n^10 +5544000*n^9 +12474000*n^8 +14196600*n^7 +6237000*n^6 -831600*n^5 +1108800*n^3 -172800*n)/11!.

G.f.: x*(x^8+232*x^7+5158*x^6+27664*x^5+47290*x^4+27664*x^3+5158*x^2+232*x+1) / (x-1)^12. - Colin Barker, May 02 2014

Formula edited by Colin Barker, May 02 2014

A085441 a(n) = Sum_{i=1..n} binomial(i+1,2)^6.

Original entry on oeis.org

1, 730, 47386, 1047386, 12438011, 98204132, 580094436, 2756876772, 11060642397, 38741283022, 121395233038, 346594833742, 914464085783, 2254559726408, 5240543726408, 11568062614344, 24395756421273, 49397866465794, 96443747465794, 182209868465794

Offset: 1

André F. Labossière, Jul 07 2003

			a(5) = C(7,3)*[191*106 + 450*(18*C(14,10) + 3851*C(13,10) + 61839*C(12,10) + 225352*C(11,10) + 225352*C(10,10))]/10010 = 12438011.

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Column k=6 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085440, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030, A234253.

[(n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12): n in [1..30]]; // G. C. Greubel, Nov 22 2017

f:= sum(binomial(1+i,2)^6,i=1..n):
seq(f, n=1..30); # Robert Israel, Nov 22 2017

Table[Sum[Binomial[i+1,2]^6,{i,n}],{n,20}] (* or *) LinearRecurrence[ {14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,730,47386,1047386,12438011, 98204132,580094436, 2756876772,11060642397, 38741283022,121395233038, 346594833742, 914464085783, 2254559726408},20] (* Harvey P. Dale, Jun 05 2017 *)

for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^6), ", ")) \\ G. C. Greubel, Nov 22 2017

G.f.: x*(x^10 +716*x^9 +37257*x^8 +450048*x^7 +1822014*x^6 +2864328*x^5 +1822014*x^4 +450048*x^3 +37257*x^2 +716*x +1) / (x -1)^14. - Colin Barker, May 02 2014

a(n) = (n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12). - G. C. Greubel, Nov 22 2017

A154286 a(n) = E(k)C(n+k,k) = Euler(k)binomial(n+k,k) for k=4.

Original entry on oeis.org

5, 25, 75, 175, 350, 630, 1050, 1650, 2475, 3575, 5005, 6825, 9100, 11900, 15300, 19380, 24225, 29925, 36575, 44275, 53130, 63250, 74750, 87750, 102375, 118755, 137025, 157325, 179800, 204600, 231880, 261800, 294525, 330225, 369075, 411255

Offset: 0

Peter Luschny, Jan 06 2009

a(n) = E(4)*binomial(n+4,4) where E(n) are the Euler number in the enumeration A122045.
a(n) is the special case k=4 in the sequence of diagonals in the triangle A153641.
a(n) is the 5th row in A093375.
a(n) is the 5th column in A103406.
a(n) is the 5th antidiagonal in A103283.
(a(n+1) - a(n))/5 are the pyramidal numbers A000292 (n>1).
(a(n+2) - 2a(n+1) + a(n))/5 are the triangular numbers A000217 (n>2).
(a(n+3) - 3a(n+2) + 3a(n+1) - a(n))/5 are the natural numbers A000027 (n > 3).
Number of orbits of Aut(Z^7) as function of the infinity norm (n+4) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 107520. - Philippe A.J.G. Chevalier, Dec 28 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A153641, A000579.

[(n+1)*(n+2)*(n+3)*(n+4)*5 div 24: n in [0..40]]; // Vincenzo Librandi, Sep 10 2016

Maple
```
seq(euler(4)*binomial(n+4,4),n=0..32);
```

CoefficientList[Series[-5/(x - 1)^5, {x, 0, 35}], x] (* Robert G. Wilson v, Jan 29 2015 *)
Table[(n + 1)*(n + 2)*(n + 3)*(n + 4)*5/24, {n, 0, 25}] (* G. C. Greubel, Sep 09 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{5,25,75,175,350},40] (* Harvey P. Dale, Nov 18 2021 *)

x='x+O('x^99); Vec(5/(1-x)^5) \\ Altug Alkan, Sep 10 2016

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*5/24.
a(n) = a(n-1)*(n+4)/n (n>0), a(0)=5.
O.g.f.: 5/(1-x)^5.
E.g.f.: (5/24)*x*(24 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Sep 09 2016
a(n) = 5*A000332(n+4). - Michel Marcus, Sep 10 2016

A001769 Expansion of 1/((1+x)*(1-x)^7).

Original entry on oeis.org

1, 6, 22, 62, 148, 314, 610, 1106, 1897, 3108, 4900, 7476, 11088, 16044, 22716, 31548, 43065, 57882, 76714, 100386, 129844, 166166, 210574, 264446, 329329, 406952, 499240, 608328, 736576, 886584, 1061208, 1263576, 1497105, 1765518, 2072862, 2423526, 2822260, 3274194, 3784858

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A002620, A002623, A001752, A001753 (first differences), A158454 (signed column k=3), A001779 (partial sums), A169794 (binomial transf.).

[(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n+5715)/5760+(-1)^n/128: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

CoefficientList[Series[1/((1+x)(1-x)^7),{x,0,30}],x] (* or *) LinearRecurrence[ {6,-14,14,0,-14,14,-6,1},{1,6,22,62,148,314,610,1106},40] (* Harvey P. Dale, May 24 2015 *)

a(n)=(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n)\/5760+1 \\ Charles R Greathouse IV, Apr 17 2012

From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+6, 6).
a(n) = (4*n^6 +96*n^5 +910*n^4 +4320*n^3 +10696*n^2 +12864*n+5715)/5760+(-1)^n/128. (End)
Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (7 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A046521 (here for the unsigned column k = 3 with offset 0).
a(n)+a(n+1) = A000579(n+7). - R. J. Mathar, Jan 06 2021

A062337 Primes whose sum of digits is 7.

Original entry on oeis.org

7, 43, 61, 151, 223, 241, 313, 331, 421, 601, 1033, 1051, 1123, 1213, 1231, 1303, 1321, 2113, 2131, 2203, 2221, 2311, 3121, 3301, 4003, 4021, 4111, 4201, 5011, 5101, 10141, 10303, 10321, 10501, 11113, 11131, 11311, 12211, 12301, 13003, 14011, 20023, 20113

Offset: 1

Amarnath Murthy, Jun 21 2001

There are O((log n)^6) members of this sequence below n.

			601 is a prime with sum of the digits = 7, hence belongs to the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A062336. See also A000579, A118703 (no digit 0)

Cf. similar sequences listed in A244918.

[p: p in PrimesUpTo(250000) | &+Intseq(p) eq 7]; // Vincenzo Librandi, Jul 08 2014

A062337 := proc(n)
    option remember ;
    local p ;
    if n = 1 then
        7;
    else
        p := nextprime(procname(n-1)) ;
        while true do
            if digsum(p) = 7 then # digsum in oeis.org/transforms.txt
                return p;
            else
                p := nextprime(p) ;
            end if;
        end do:
    end if;
end proc:
seq(A062337(n),n=1..80) ; # R. J. Mathar, May 22 2025

Select[Prime[Range[3000]], Plus @@ IntegerDigits[ # ] == 7 &] (* Zak Seidov, Feb 17 2005 *)

A062337(lim)={my(pow=ceil(log(floor(lim)-.5)/log(10)),n);print("Checking for members of A062337 up to 10^"pow);for(a=0,pow-1,for(b=0,a,for(c=0,b,for(d=0,c,for(e=0,d,for(f=0,e,n=10^a+10^b+10^c+10^d+10^e+10^f+1;if(isprime(n),print1(n","))))))))};

select( {is_A062337(p, s=7)=sumdigits(p)==s&&isprime(p)}, primes([1, 14321])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

{A062337_upto_length(L, s=7, a=List(), u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1, L]|i<-[1..s]], isprime(p=vecsum(vecextract(u, d))) && listput(a, p), 1); Vecrev(a)} \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A052221 (digit sum 7). - M. F. Hasler, Mar 09 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Comments and program from Charles R Greathouse IV, Sep 11 2009

A087107 This table shows the coefficients of combinatorial formulas needed for generating the sequential sums of p-th powers of tetrahedral numbers. The p-th row (p>=1) contains a(i,p) for i=1 to 3p-2, where a(i,p) satisfies Sum_{i=1..n} C(i+2,3)^p = 4 C(n+3,4) * Sum_{i=1..3p-2} a(i,p) C(n-1,i-1)/(i+3).

Original entry on oeis.org

1, 1, 3, 3, 1, 1, 15, 69, 147, 162, 90, 20, 1, 63, 873, 5191, 16620, 31560, 36750, 25830, 10080, 1680, 1, 255, 9489, 130767, 919602, 3832650, 10238000, 18244380, 21990360, 17745000, 9198000, 2772000, 369600, 1, 1023, 97953, 2903071, 40317780

Offset: 1

André F. Labossière, Aug 11 2003

Let s_n denote the sequence (1, 4^n, 10^n, 20^n, ...) regarded as an infinite column vector, where 1, 4, 10, 20, ... is the sequence of tetrahedral numbers A000292. It appears that the n-th row of this table is determined by the matrix product P^(-1)s_n, where P denotes Pascal's triangle A007318. - Peter Bala, Nov 26 2017
From Peter Bala, Mar 11 2018: (Start)
The observation above is correct.
The table entries T(n,k) are the coefficients when expressing the polynomial C(x+3,3)^p of degree 3*p in terms of falling factorials: C(x+3,3)^p = Sum_{k = 0..3*p} T(p,k)*C(x,k). It follows that Sum_{i = 0..n-1} C(i+3,3)^p = Sum_{k = 0..3*p} T(p,k)*C(n,k+1).
The sum of the p-th powers of the tetrahedral numbers is also given by Sum_{i = 0..n-1} C(i+3,3)^p = Sum_{k = 3..3*p} A299041(p,k)*C(n+3,k+1) for p >= 1. (End)

			Row 3 contains 1,15,69,147,162,90,20, so Sum_{i=1..n} C(i+2,3)^3 = 4 * C(n+3,4) * [ a(1,3)/4 + a(2,3)*C(n-1,1)/5 + a(3,3)*C(n-1,2)/6 + ... + a(7,3)*C(n-1,6)/10 ] = 4 * C(n+3,4) * [ 1/4 + 15*C(n-1,1)/5 + 69*C(n-1,2)/6 + 147*C(n-1,3)/7 + 162*C(n-1,4)/8 + 90*C(n-1,5)/9 + 20*C(n-1,6)/10 ]. Cf. A086021 for more details.
From _Peter Bala_, Mar 11 2018: (Start)
Table begins
n=0 | 1
n=1 | 1  3   3    1
n=2 | 1 15  69  147   162    90    20
n=3 | 1 63 873 5191 16620 31560 36750 25830 10080 1680
...
Row 2: C(i+3,3)^2 = C(i,0) + 15*C(i,1) + 69*C(i,2) + 147*C(i,3) + 162*C(i,4) + 90*C(i,5) + 20*C(i,6). Hence, Sum_{i = 0..n-1} C(i+3,3)^2 =  C(n,1) + 15*C(n,2) + 69*C(n,3) + 147*C(n,4) + 162*C(n,5) + 90*C(n,6) + 20*C(n,7). (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A000292, A024166, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A087108, A000389, A086023, A086024, A087109, A000579, A086025, A086026, A087110, A000580, A086027, A086028, A087111, A027555, A086029, A086030.

Cf. A087127, A087110, A174266, A299041.

seq(seq(add( (-1)^(k-i)*binomial(k, i)*binomial(i+3, 3)^n, i= 0..k), k = 0..3*n), n = 0..8); # Peter Bala, Mar 11 2018

a[i_, p_] := Sum[Binomial[i - 1, 2*k - 2]*Binomial[i - 2*k + 4, i - 2*k + 1]^(p - 1) - Binomial[i - 1, 2*k - 1]*Binomial[i - 2*k + 3, i - 2*k]^(p - 1), {k, 1, (2*i + 1 + (-1)^(i - 1))/4}]; Table[If[p == 1, 1, a[i, p]], {p, 1, 10}, {i, 1, 3*p - 2}]//Flatten (* G. C. Greubel, Nov 23 2017 *)

{a(i, p) = sum(k=1, (2*i + 1 + (-1)^(i - 1))/4, binomial(i - 1, 2*k - 2)*binomial(i - 2*k + 4, i - 2*k + 1)^(p - 1) - binomial(i - 1, 2*k - 1)*binomial(i - 2*k + 3, i - 2*k)^(p - 1))}; for(p=1,8, for(i=1, 3*p-2, print1(if(p==1,1,a(i,p)), ", "))) \\ G. C. Greubel, Nov 23 2017

a(i, p) = Sum_{k=1..[2*i+1+(-1)^(i-1)]/4} [ C(i-1, 2*k-2)*C(i-2*k+4, i-2*k+1)^(p-1) -C(i-1, 2*k-1)*C(i-2*k+3, i-2*k)^(p-1) ].
From Peter Bala, Nov 26 2017: (Start)
Conjectural formula for table entries: T(n,k) = Sum_{j = 0..k} (-1)^(k+j)*binomial(k,j)*binomial(j+3,3)^n.
Conjecturally, the n-th row polynomial R(n,x) = 1/(1 + x)*Sum_{i >= 0} binomial(i+3,3)^n *(x/(1 + x))^n. (End)
From Peter Bala, Mar 11 2018: (Start)
The conjectures above are correct.
The following remarks assume the row and column indices start at 0.
T(n+1,k) = C(k+3,3)*T(n,k) + 3*C(k+2,3)*T(n,k-1) + 3*C(k+1,3)*T(n,k-2) + C(k,3)*T(n,k-3) with boundary conditions T(n,0) = 1 for all n and T(n,k) = 0 for k > 3*n.
Sum_{k = 0..3*n} T(n,k)*binomial(x,k) = (binomial(x+3,3))^n.
x^3*R(n,x) = (1 + x)^3 * the n-th row polynomial of A299041.
R(n+1,x) = 1/3!*(1 + x)^3*(d/dx)^3 (x^3*R(n,x)).
(1 - x)^(3*n)*R(n,x/(1 - x)) gives the n-th row polynomial of A174266.
R(n,x) = (1 + x)^3 o (1 + x)^3 o ... o (1 + x)^3 (n factors), where o denotes the black diamond product of power series defined in Dukes and White. Note the polynomial x^3 o ... o x^3 (n factors) is the n-th row polynomial of A299041. (End)

Edited by Dean Hickerson, Aug 16 2003

A087111 This table shows the coefficients of combinatorial formulas needed for generating the sequential sums of p-th powers of binomial coefficients C(n,7). The p-th row (p>=1) contains a(i,p) for i=1 to 7p-6, where a(i,p) satisfies Sum_{i=1..n} C(i+6,7)^p = 8 C(n+7,8) * Sum_{i=1..7p-6} a(i,p) C(n-1,i-1)/(i+7).

Original entry on oeis.org

1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 63, 1169, 10703, 58821, 214123, 545629, 1004307, 1356194, 1347318, 974862, 500346, 172788, 36036, 3432, 1, 511, 45633, 1589567, 29302889, 333924087, 2577462937, 14287393351, 59159005164, 188008120188

Offset: 1

André F. Labossière, Aug 11 2003

From Peter Bala, Mar 11 2018: (Start)

The table entries T(n,k) are the coefficients when expressing the polynomial C(x+7,7)^p of degree 7*p in terms of falling factorials: C(x+7,7)^p = Sum_{k = 0..7*p} T(p,k)*C(x,k). It follows that Sum_{i = 0..n-1} C(i+7,7)^p = Sum_{k = 0..7*p} T(p,k)*C(n,k+1). (End)

			Row 3 contains 1,63,1169,...,3432, so Sum_{i=1..n} C(i+6,7)^3 = 8 * C(n+7,8) * [ a(1,3)/8 + a(2,3)*C(n-1,1)/9 + a(3,3)*C(n-1,2)/10 + ... + a(15,3)*C(n-1,14)/22 ] = 8 * C(n+7,8) * [ 1/8 + 63*C(n-1,1)/9 + 1169*C(n-1,2)/10 + ... + 3432*C(n-1,14)/22 ]. Cf. A086030 for more details.

G. C. Greubel, Table of n, a(n) for the first 40 rows, flattebed
Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A000292, A024166, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A087107, A000332, A086020, A086021, A086022, A087108, A000389, A086023, A086024, A087109, A000579, A086025, A086026, A087110, A000580, A086027, A086028, A027555, A086029, A086030, A087127.

seq(seq(add( (-1)^(k-i)*binomial(k, i)*binomial(i+7, 7)^n, i = 0..k), k = 0..7*n), n = 0..4); # Peter Bala, Mar 11 2018

a[i_, p_] := Sum[Binomial[i - 1, 2*k - 2]*Binomial[i - 2*k + 8, i - 2*k + 1]^(p - 1) - Binomial[i - 1, 2*k - 1]*Binomial[i - 2*k + 7, i - 2*k]^(p - 1), {k, 1, (2*i + 1 + (-1)^(i - 1))/4}]; Table[If[p == 1, 1, a[i, p]], {p, 1, 10}, {i, 1, 7*p - 6}]//Flatten (* G. C. Greubel, Nov 23 2017 *)

{a(i, p) = sum(k=1, (2*i + 1 + (-1)^(i - 1))/4, binomial(i - 1, 2*k - 2)*binomial(i - 2*k + 8, i - 2*k + 1)^(p - 1) - binomial(i - 1, 2*k - 1)*binomial(i - 2*k + 7, i - 2*k)^(p - 1))}; for(p=1,8, for(i=1, 7*p-6, print1(if(p==1,1,a(i,p)), ", "))) \\ G. C. Greubel, Nov 23 2017

a(i, p) = Sum_{k=1..[2*i+1+(-1)^(i-1)]/4} [ C(i-1, 2*k-2)*C(i-2*k+8, i-2*k+1)^(p-1) -C(i-1, 2*k-1)*C(i-2*k+7, i-2*k)^(p-1) ]
From Peter Bala, Mar 11 2018: (Start)
The following remarks assume the row and column indices start at 0.
T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i) * binomial(i+7,7)^n. Equivalently, let v_n denote the sequence (1, 8^n, 36^n, 120^n, ...) regarded as an infinite column vector, where 1, 8, 36, 120, ... is the sequence binomial(n+7,7) - see A000580. Then the n-th row of this table is determined by the matrix product P^(-1)*v_n, where P denotes Pascal's triangle A007318.
Recurrence: T(n+1,k) = Sum_{i = 0..7} C(7,i)*C(k+7-i,7)*T(n,k-i) with boundary conditions T(n,0) = 1 for all n and T(n,k) = 0 for k > 7*n.
n-th row polynomial R(n,x) = (1 + x)^7 o (1 + x)^7 o ... o (1 + x)^7 (n factors), where o denotes the black diamond product of power series defined in Dukes and White.
R(n+1,x) = 1/7!*(1 + x)^7 * (d/dx)^7(x^7*R(n,x)).
R(n,x) = Sum_{i >= 0} binomial(i+7,7)^n*x^i/(1 + x)^(i+1).
(End)

Edited by Dean Hickerson, Aug 16 2003

cp's OEIS Frontend

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

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A035038 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,5).

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A085439 a(n) = Sum_{i=1..n} binomial(i+1,2)^4.

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A085440 a(n) = Sum_{i=1..n} binomial(i+1,2)^5.

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A085441 a(n) = Sum_{i=1..n} binomial(i+1,2)^6.

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A154286 a(n) = E(k)*C(n+k,k) = Euler(k)*binomial(n+k,k) for k=4.

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A154286 a(n) = E(k)C(n+k,k) = Euler(k)binomial(n+k,k) for k=4.