A005585 -id:A005585 - cp's OEIS frontend

A101095 Fourth difference of fifth powers (A000584).

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

A107891 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(3n^2 + 15n + 20)/2880.

Original entry on oeis.org

1, 19, 155, 805, 3136, 9996, 27468, 67320, 150645, 313027, 611611, 1134497, 2012920, 3436720, 5673648, 9093096, 14194881, 21643755, 32310355, 47319349, 68105576, 96479020, 134699500, 185562000, 252493605, 339663051, 452103939

Offset: 0

Emeric Deutsch, Jun 12 2005

Kekulé numbers for certain benzenoids.

Partial sums of A114239. First differences of A047819. - Peter Bala, Sep 21 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 167, 187 and p. 105 eq. (iii) for k=2 and m=5).

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A005585, A006542, A047819, A114239, A114242.

a:=n->(1/2880)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n^2+15*n+20): seq(a(n),n=0..32);

Table[((1+n) (2+n)^2 (3+n)^2 (4+n) (20+3 n (5+n)))/2880,{n,0,40}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,19,155,805,3136,9996,27468,67320,150645},40] (* Harvey P. Dale, Dec 10 2021 *)

a(n-2) = (1/8) * Sum_{1 <= x_1, x_2 <= n} (x_1*x_2)^2*(det V(x_1,x_2))^2 = 1/8*sum {1 <= i,j <= n} (i*j*(i-j))^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^9. - Colin Barker, Feb 08 2012
a(n) = (A000330(n+2)*A000538(n+2) - (A000537(n+2))^2)/4. - J. M. Bergot, Sep 17 2013
Sum_{n>=0} 1/a(n) = 17095/4 - 240*Pi^2 - 162*sqrt(15)*Pi*tanh(sqrt(5/3)*Pi/2). - Amiram Eldar, May 29 2022

A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A002418
Antidiagonal sums:  A005585
row 1,  (1,3,6,...)**(1,2,3,...):  A000332
row 2,  (1,3,6,...)**(2,3,4,...):  A005582
row 3,  (1,3,6,...)**(3,4,5,...):  A095661
row 4,  (1,3,6,...)**(4,5,6,...):  A095667
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15...35....70....126
2....9....25...55....105...182
3....13...35...75....140...238
4....17...45...95....175...294
5....21...55...115...210...350

Cf. A213500, A213548.

b[n_] := n (n + 1)/2; 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}]  (* A213550 *)
d = Table[t[n, n], {n, 1, 40}] (* A002418 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

A114242 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n+5)/720.

Original entry on oeis.org

1, 14, 90, 385, 1274, 3528, 8568, 18810, 38115, 72358, 130130, 223587, 369460, 590240, 915552, 1383732, 2043621, 2956590, 4198810, 5863781, 8065134, 10939720, 14651000, 19392750, 25393095, 32918886, 42280434, 53836615, 68000360

Offset: 0

Emeric Deutsch, Nov 18 2005

Kekulé numbers for certain benzenoids.

Partial sums of A114244. First differences of A006857. - Peter Bala, Sep 21 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/2 and p. 105, eq. (ii) K(Ob(2,4,n))).

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005585, A006542, A107891.

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720: seq(a(n),n=0..30);

Table[((n+1)(n+2)^2 (n+3)^2 (n+4)(2n+5))/720,{n,0,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,14,90,385,1274,3528,8568,18810},30] (* Harvey P. Dale, Aug 21 2013 *)

a(n)=(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+x)(1 + 5x + x^2)/(1-x)^8.
a(n-2) = (1/6) * Sum_{1 <= x_1, x_2 <= n} (x_1)^2*x_2*(det V(x_1,x_2))^2 = (1/6)*Sum_{1 <= i,j <= n} i^2*j*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Aug 21 2013
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 3550 - 5120*log(2).
Sum_{n>=0} (-1)^n/a(n) = 3430 - 1280*Pi + 60*Pi^2. (End)

A213564 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 4, 27, 21, 9, 77, 67, 43, 16, 182, 167, 127, 73, 25, 378, 357, 297, 207, 111, 36, 714, 686, 602, 467, 307, 157, 49, 1254, 1218, 1106, 917, 677, 427, 211, 64, 2079, 2034, 1890, 1638, 1302, 927, 567, 273, 81, 3289, 3234, 3054, 2730, 2282, 1757

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213565
Antidiagonal sums:  A101094
Row 1,  (1,3,6,...)**(1,4,9,...):  A005585
Row 2,  (1,3,6,...)**(4,9,16,...):  (k^5 +25*k^4 + 60*k^3 + 215*k^2 + 59*k)/60
Row 3,  (1,3,6,...)**(9,16,25,...):  (k^5 +35*k^4 + 30*k^3 + 505*k^2 + 149*k)/60
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
4....21....67....167...357
9....43....127...297...602
16...73....207...467...917
25...111...307...677...1302
36...157...427...927...1757

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

Cf. A213500.

b[n_] := n (n + 1)/2; c[n_] := n^2
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}]  (* A213564 *)
d = Table[t[n, n], {n, 1, 40}] (* A213565 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101094 *)

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.

A081284 An interleaved sequence of pyramidal and polygonal numbers.

Original entry on oeis.org

0, 1, 1, 6, 7, 22, 27, 62, 77, 147, 182, 308, 378, 588, 714, 1044, 1254, 1749, 2079, 2794, 3289, 4290, 5005, 6370, 7371, 9191, 10556, 12936, 14756, 17816, 20196, 24072, 27132, 31977, 35853, 41838, 46683, 53998, 59983, 68838, 76153, 86779, 95634

Offset: 0

Paul Barry, Mar 17 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Cf. A081283, A081282.

CoefficientList[Series[x (1 + x^3) / ((1 - x) (1 - x^2)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2013 *)

G.f.: x*(1+x^3)/((1-x)*(1-x^2)^5).
a(2*n) = A005585(n); a(2*n+1) = A081282(n+1).
G.f.: (x*(x^2-x+1))/((x+1)^4*(x-1)^6). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n+9-(-1)^n)*(2*n^2+15*n+51+3*(n-5)*(-1)^n)/30720. - Luce ETIENNE, Mar 13 2015

A121306 Array read by antidiagonals: a(m,n) = a(m,n-1)+a(m-1,n) but with initialization values a(0,0)=0, a(m>=1,0)=1, a(0,1)=1, a(0,n>1)=0.

Original entry on oeis.org

2, 2, 3, 2, 5, 4, 2, 7, 9, 5, 2, 9, 16, 14, 6, 2, 11, 25, 30, 20, 7, 2, 13, 36, 55, 50, 27, 8, 2, 15, 49, 91, 105, 77, 35, 9, 2, 17, 64, 140, 196, 182, 112, 44, 10, 19, 81, 204, 336, 378, 294, 156, 54, 100, 285, 540, 714, 672, 450, 210, 385, 825, 1254, 1386, 1122

Offset: 0

Thomas Wieder, Aug 04 2006, Aug 06 2006

For a(1,0)=1, a(m>1,0)=0 and a(0,n>=0)=0 one gets Pascal's triangle A007318.

			Array begins
2 2 2 2 2 2 2 2 2 ...
3 5 7 9 11 13 15 17 19 ...
4 9 16 25 36 49 64 81 100 ...
5 14 30 55 91 140 204 285 385 ...
6 20 50 105 196 336 540 825 1210 ...
7 27 77 182 378 714 1254 2079 3289 ...

The first nine rows are: A006527, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347.
The initial columns are: A000027, A000096, A005581, A005582, A005583, A005584.
Cf. A119800, A007318, A006527, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A000027, A000096, A005581, A005582, A005583, A005584.

=Z(-1)S+ZS(-1). The very first row (not included into the table) contains the initialization values: a(0,1)=1, a(0,n>=2)=0. The very first column (not included into the table) contains the initialization values: a(m>=1,0)=1. The value a(0,0)=0 does not enter into the table.

a(m,n) = a(m,n-1)+a(m-1,n), a(0,0)=0, a(m>=1,0)=1, a(0,1)=1, a(0,n>1)=0.

Edited by N. J. A. Sloane, Sep 15 2006

A191662 a(n) = n! / A000034(n-1).

Original entry on oeis.org

1, 1, 6, 12, 120, 360, 5040, 20160, 362880, 1814400, 39916800, 239500800, 6227020800, 43589145600, 1307674368000, 10461394944000, 355687428096000, 3201186852864000, 121645100408832000, 1216451004088320000, 51090942171709440000, 562000363888803840000

Offset: 1

Paul Curtz, Jun 10 2011

The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
A005408 = (2*n+1)/1                       = 1, 3,  5,   7,   9, ... (k=1)
A000290 = (n^2)/1                         = 1, 4,  9,  16,  25, ... (k=2)
A000330 = n*(n+1)*(2*n+1)/6               = 1, 5, 14,  30,  55, ... (k=3)
A002415 = (n^2)*(n^2-1)/12                = 1, 6, 20,  50, 105, ... (k=4)
A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27,  77, 182, ... (k=5)
A040977 = (n^2)*(n^2-1)*(n^2-4)/360       = 1, 8, 35, 112, 294, ... (k=6)
A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
The first superdiagonal of this array appears in A029651. - Paul Curtz, Jul 04 2011
The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - Johannes W. Meijer, Jun 22 2011

Cf. A000142, A009445, A002674, A064680.
Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
Cf. A052612.

A191662:= proc(n): n!/A000034(n-1) end: A000034 := proc(n) op((n mod 2)+1, [1, 2]) ; end proc: seq(A191662(n),n=1..17); # Johannes W. Meijer, Jun 22 2011

Array[If[EvenQ[#],#!/2,#!]&,20] (* Harvey P. Dale, Mar 14 2014 *)

a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
a(n+1) = A064680(n+1) * a(n).
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - Georg Fischer, Nov 25 2022
a(n) = A052612(n)/2 for n >= 1. - Alois P. Heinz, Sep 05 2023

More terms from Harvey P. Dale, Mar 14 2014

A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

Original entry on oeis.org

1, 3, 5, 1, 7, 5, 9, 14, 1, 11, 30, 7, 13, 55, 27, 1, 15, 91, 77, 9, 17, 140, 182, 44, 1, 19, 204, 378, 156, 11, 21, 285, 714, 450, 65, 1, 23, 385, 1254, 1122, 275, 13, 25, 506, 2079, 2508, 935, 90, 1, 27, 650, 3289, 5148, 2717, 442, 15, 29, 819, 5005, 9867

Offset: 1

Clark Kimberling, Feb 27 2012

			First five rows:
  1
  3
  5    1
  7    5
  9   14   1
First five polynomials v(n,x):
  1
  3
  5 +   x
  7 +  5x
  9 + 14x + x^2

Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.
Row sums, v(n,1): A003948.
Alternating row sums, v(n,-1): A090131.
Cf. A208508.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.

Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - Knud Werner, Jan 11 2022

A211235 Array of generalized Eulerian numbers C(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 7, 10, 4, 1, 12, 27, 20, 5, 1, 21, 69, 77, 35, 6, 1, 38, 176, 272, 182, 56, 7, 1, 71, 456, 936, 846, 378, 84, 8, 1, 136, 1205, 3210, 3750, 2232, 714, 120, 9, 1, 265, 3247, 11075, 16290, 12342, 5214, 1254, 165, 10

Offset: 1

N. J. A. Sloane, Apr 05 2012

			Array begins:
  1,       2,   3,   4,    5,     6, ... A000027
  1,       4,  10,  20,   35,    56, ... A000292
  1,       7,  27,  77,  182,   378, ... A005585
  1,      12,  69, 272,  846,  2232, ... A101097
  1,      21, 176, 936, 3750, 12342, ... A254681
     A005126,
  ...
Triangle begins:
  1
  1  2
  1  4   3
  1  7  10   4
  1 12  27  20   5
  1 21  69  77  35  6
  1 38 176 272 182 56 7
  ...

D. H. Lehmer, Generalized Eulerian numbers, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).

Cf. A008292, A211232-A211235.

A211235 := (n, k) -> add(binomial(n-i, k-i)*i^(n-k), i = 1 .. k): for n from 1 to 10 do seq(A211235(n, k), k = 1 .. n) end do; # Peter Bala, Oct 27 2015

T[n_, k_] := Sum[Binomial[n-i, k-i] * i^(n-k), {i, 1, k}]; Table[T[n, k], {n,1,10}, {k,1,n}] //Flatten (* Amiram Eldar, Nov 30 2018 *)

From Peter Bala, Oct 27 2015: (Start)
O.g.f. of n-th row of square array: 1/(1 - x)^n * (x*d/dx)^n log(1/(1 - x)), for n >= 1.
E.g.f. of square array: log((1 - x)/(1 - x*exp(t/(1 - x)))).
Read as a triangle: T(n,k) = Sum_{i = 1..k} binomial(n-i,k-i)*i^(n-k) for 1 <= k <= n.
n-th row polynomial of triangle: Sum_{i = 0..n-1} x^i*(x + i)^(n-i). (End)

Terms a(37)-a(55) added by Peter Bala, Oct 27 2015

cp's OEIS Frontend

A101095 Fourth difference of fifth powers (A000584).

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A107891 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.

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A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A114242 a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(2n+5)/720.

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A213564 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

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A081284 An interleaved sequence of pyramidal and polygonal numbers.

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A121306 Array read by antidiagonals: a(m,n) = a(m,n-1)+a(m-1,n) but with initialization values a(0,0)=0, a(m>=1,0)=1, a(0,1)=1, a(0,n>1)=0.

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A107891 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(3n^2 + 15n + 20)/2880.

A114242 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n+5)/720.