A142978 -id:A142978 - cp's OEIS frontend

A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 19, 8, 1, 6, 25, 44, 33, 10, 1, 7, 36, 85, 96, 51, 12, 1, 8, 49, 146, 225, 180, 73, 14, 1, 9, 64, 231, 456, 501, 304, 99, 16, 1, 10, 81, 344, 833, 1182, 985, 476, 129, 18, 1, 11, 100, 489, 1408, 2471, 2668, 1765, 704, 163, 20, 1, 12

Offset: 0

Gary W. Adamson, Mar 19 2005

The n-th column of the triangle is the binomial transform of the n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). A104698 = reversal by rows of A142978. - Gary W. Adamson, Jul 17 2008
This sequence is jointly generated with A210222 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) + 1 and v(n,x) = 2x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section at A210222. - Clark Kimberling, Mar 19 2012
This Riordan triangle T appears in a formula for A001100(n, 0) = A002464(n), for n >= 1. - Wolfdieter Lang, May 13 2025

			The Riordan triangle T begins:
  n\k  0   1   2    3    4    5    6   7   8  9 10 ...
  ----------------------------------------------------
  0:   1
  1:   2   1
  2:   3   4   1
  3:   4   9   6    1
  4:   5  16  19    8    1
  5:   6  25  44   33   10    1
  6:   7  36  85   96   51   12    1
  7:   8  49 146  225  180   73   14   1
  8:   9  64 231  456  501  304   99  16   1
  9:  10  81 344  833 1182  985  476 129  18  1
  10: 11 100 489 1408 2471 2668 1765 704 163 20  1
  ... reformatted and extended by _Wolfdieter Lang_, May 13 2025
From _Wolfdieter Lang_, May 13 2025: (Start)
Zumkeller recurrence (adapted for offset [0,0]): 19 = T(4, 2) = T(2, 1) + T(3, 1) + T(3,3) = 4 + 9 + 6 = 19.
A-sequence recurrence: 19 = T(4, 2) = 1*T(3. 1) + 2*T(3. 2) - 2*T(3, 3) = 9 + 12 - 2 = 19.
Z-sequence recurrence: 5 = T(4, 0) = 2*T(3, 0) - 1*T(3, 1) + 2*T(3, 2) - 6*T(3, 3) = 8 - 9 + 12 + 6 = 5.
Boas-Buck recurrence: 19 = T(4, 2) = (1/2)*((2 + 0)*T(2, 2) + (2 + 2*2)*T(3, 2)) = (1/2)*(2 + 36) = 19. (End)

Reinhard Zumkeller, First 100 rows of triangle, flattened

Diagonal sums are A008937(n+1).
Cf. A048739 (row sums), A008288, A005900 (column 3), A014820 (column 4)
Cf. A081277, A142978 by antidiagonals, A119328, A110271 (matrix inverse).
Cf. A001100, A002464, A010673, A040000, A112478, A133872, A383857.

a104698 n k = a104698_tabl !! (n-1) !! (k-1)
a104698_row n = a104698_tabl !! (n-1)
a104698_tabl = [1] : [2,1] : f [1] [2,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) ([1] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Jul 17 2015

A104698 := proc(n, k) add(binomial(k, j)*binomial(n-j+1, n-k-j), j=0..n-k) ; end proc:
seq(seq(A104698(n, k), k=0..n), n=0..15); # R. J. Mathar, Sep 04 2011
T := (n, k) -> binomial(n + 1, k + 1)*hypergeom([-k, k - n], [-n - 1], -1):
for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
T := proc(n, k) option remember; if k = 0 then n + 1 elif k = n then 1 else T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) fi end: # Peter Luschny, May 13 2025

u[1, ] = 1; v[1, ] = 1;
u[n_, x_] := u[n, x] = x u[n-1, x] + v[n-1, x] + 1;
v[n_, x_] := v[n, x] = 2 x u[n-1, x] + v[n-1, x] + 1;
Table[CoefficientList[u[n, x], x], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 10 2019, after Clark Kimberling *)

T(n,k)=sum(j=0,n-k,binomial(k,j)*binomial(n-j+1,k+1)) \\ Charles R Greathouse IV, Jan 16 2012

The triangle is extracted from the product A * B; A = [1; 1, 1; 1, 1, 1; ...], B = [1; 1, 1; 1, 3, 1; 1, 5, 5, 1; ...] both infinite lower triangular matrices (rest of the terms are zeros). The triangle of matrix B by rows = A008288, Delannoy numbers.
From Paul Barry, Jul 18 2005: (Start)
Riordan array (1/(1-x)^2, x(1+x)/(1-x)) = (1/(1-x), x)*(1/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{j=0..n} Sum_{i=0..j-k} C(j-k, i)*C(k, i)*2^i.
T(n, k) = Sum_{j=0..k} Sum_{i=0..n-k-j} (n-k-j-i+1)*C(k, j)*C(k+i-1, i). (End)
T(n, k) = binomial(n+1, k+1)*2F1([-k, k-n], [-n-1], -1) where 2F1 is a Gaussian hypergeometric function. - R. J. Mathar, Sep 04 2011
T(n, k) = T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) for 1 < k < n; T(n, 0) = n + 1; T(n, n) = 1. - Reinhard Zumkeller, Jul 17 2015
From Wolfdieter Lang, May 13 2025: (Start)
The Riordan triangle T = (1/(1 - x)^2, x*(1 + x)/(1 - x)) has the o.g.f. G(x, y) = 1/((1 - x)*(1 - x - y*x*(1+x))) for the row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k.
The o.g.f. for column k is G(k, x) = (1/(1 - x)^2)*(x*(1 + x)/(1 - x))^k, for k >= 0.
The o.g.f. for the diagonal m is D(m, x) = N(m, x)/(1 - x)^(m+1), with the numerator polynomial N(m, x) = Sum_{k=0..floor(m/2)} A034867(m, k)*x^(2*k) for m >= 0.
The row sums with o.g.f. R(x) = 1/((1 -x)*(1 - 2*x -x^2) give A048739.
The alternating row sums with o.g.f. 1/((1 - x)(1 + x^2)) give A133872.
The A-sequence for this Riordan triangle has o.g.f. A(x) = 1 + x + sqrt(1 + 6*x + x^2))/2 giving A112478(n). Hence T(n, k) = Sum_{j=0..n-k} A112478(j)*T(n-1, k-1+j), for n >= 1, k >= 1, T(n, k) = 0 for n < k, and T(0, 0) = 1.
The Z-sequence has o.g.f. (5 + x - sqrt(1 + 6*x + x^2))/2 = 3 + x - A(x) giving Z(n) = {2, -1, -A112478(n >= 2)}. Hence T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1. For A- and Z-sequences of Riordan triangles see a W. Lang link at A006232 with references.
The Boas-Buck sequences alpha and beta for the Riordan triangle T (see A046521 for the Aug 10 2017 comment and reference) are alpha(n) = A040000(n+1) = repeat{2} and beta(n) = A010673(n+1) = repeat{2,0}. Hence the recurrence for column T(n, k){n>=k}, with input T(k, k) = 1, for k >= 0, is T(n, k) = (1/(n-k)) * Sum{j=k..n-1} (2 + k*(1 + (-1)^(n-1-j))) *T(j,k), for n >= k+1. (End)

A217873 a(n) = 4n(n^2 + 2)/3.

Original entry on oeis.org

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004

Offset: 0

M. F. Hasler, Oct 13 2012

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

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. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

[4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012

Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)

makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */

PARI
```
a(n)=(n^2+2)*n/3*4
```

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138.  (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

A072410 Number of iterations of the map k -> A000001(k) needed to reach 1 starting at n, or -1 if no such number exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

The old entry with this sequence number was a duplicate of A052409.
It appears that a(n) is the number of times n appears in A142978, excluding the first column of infinitely many 1's. - Ron Wolf, Dec 16 2020
Preceding comment is incorrect. The first counterexample is a(19) = 1, whereas 19 appears twice in A142978. - Eric M. Schmidt, Mar 22 2021

			Conway et al. remark that every number less than 2048 reaches 1 after at most 5 steps and give the following examples:
672 -> 1280 -> 1116461 -> 1
1024 -> 49487367289 -> 1
720 -> 840 -> 186 -> 6 -> 2 -> 1
320 -> 1640 -> 68 -> 5 -> 1
384 -> 20169 -> 67 -> 1
128 -> 2328 -> 64 -> 267 -> 1
960 -> 11394 -> 60 -> 13 -> 1
864 -> 4725 -> 51 -> 1
1344 -> 11720 -> 49 -> 2 -> 1
1440 -> 5958 -> 16 -> 14 -> 2 -> 1
1248 -> 1460 -> 15 -> 1
256 -> 56092 -> 11 -> 1
1728 -> 47937 -> 6 -> 2 -> 1
512 -> 10494213 -> 5 -> 1
1536 -> 408641062 -> 4 -> 2 -> 1
1664 -> 21507 -> 2 -> 1
1280 -> 1116461 -> 1

Eric M. Schmidt, Table of n, a(n) for n = 1..2047 [using data from example and A000001. a(1024) corrected by _Jinyuan Wang_, Jun 26 2022]
John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Cf. A000001, A066952 (indices of records).

A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 10, 19, 4, 1, 14, 51, 44, 5, 1, 18, 99, 180, 85, 6, 1, 22, 163, 476, 501, 146, 7, 1, 26, 243, 996, 1765, 1182, 231, 8, 1, 30, 339, 1804, 4645, 5418, 2471, 344, 9, 1, 34, 451, 2964, 10165, 17718, 14407, 4712, 489, 10

Offset: 0

Peter Bala, Jul 15 2008

The row entries are the figurate numbers of the odd dimensional cross polytopes. See A142978 for the complete table of figurate numbers of n-dimensional cross polytopes. The rows are the partial sums of the even-numbered rows of the square array of Delannoy numbers A008288.

			The square array begins
 n\k| 0...1....2.....3.....4.......5
------------------------------------
 .0.| 1...2....3.....4......5......6 ... A000027
 .1.| 1...6...19....44.....85....146 ... A005900
 .2.| 1..10...51...180....501...1182 ... A069038
 .3.| 1..14...99...476...1765...5418 ... A099193
 .4.| 1..18..163...996...4645..17718 ... A099196
 .5.| 1..22..243..1804..10165..46530 ... A300624
 ...

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.

Cf. A005900 (row 1), A008288, A069038 (row 2), A099193 (row 3), A099196 (row 4), A300624 (row 5), A142978, A142983.

with(combinat): T:=(n,k) -> add(binomial(2n,k-j)*binomial(2n+j+1,j), j = 0..k): for n from 0 to 9 do seq(T(n,k), k = 0..9) end do;

T(n,k) = Sum_{j = 0..k} C(2*n, k-j)*C(2*n+j+1, j).
O.g.f.: 1/{(1 - x)^2 - y*(1 + x)^2} = Sum_{n, k >= 0} T(n,k)*x^k*y^n = 1/(1 - y) * Sum_{m >= 0} U(m, (1 + y)/(1 - y))*x^m, where U(m, y) denotes the m-th Chebyshev polynomial of the second kind.
O.g.f. row n: (1 + x)^(2*n)/(1 - x)^(2*n+2).
O.g.f. column k: 1/(1 - y)*U(k, (1 + y)/(1 - y)).
The entries in the n-th row appear in the series acceleration formula for the constant log(2): Sum_{k >= 1} (-1)^(k+1)/(T(n,k)*T(n,k+1)) = 1 + (4*n + 2)*( log(2) - (1 - 1/2 + 1/3 - ... + 1/(2*n + 1)) ).
For example, n = 1 gives log(2) = 4/6 + (1/6)*( 1/(1*6) - 1/(6*19) + 1/(19*44) - 1/(44*85) + ... ). See A142983 for further details.

Restored missing program. - Peter Bala, Oct 02 2008

A361745 Square array of circular Delannoy numbers A(i,j) (i >= 0, j >= 0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 16, 6, 1, 1, 8, 36, 36, 8, 1, 1, 10, 64, 114, 64, 10, 1, 1, 12, 100, 264, 264, 100, 12, 1, 1, 14, 144, 510, 768, 510, 144, 14, 1, 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1, 1, 18, 256, 1386, 3648, 5010, 3648, 1386, 256, 18, 1

Offset: 0

Noah Snyder, Mar 22 2023

An (n,m) Delannoy loop is an oriented unbased loop on a toroidal grid with points labeled by Z/n x Z/m composed of steps of the form (1,0), (0,1), and (1,1), and which loops around the torus exactly once in each of the x-direction and the y-direction. The circular Delannoy numbers count the number of (n,m) Delannoy loops. This array is a modification of the ordinary Delannoy numbers A008288.

Dimensions of hom spaces Hom(S^{{i}}, S^{{j}}) in the circular Delannoy category attached to the oligomorphic group of order preserving self-bijections of the circle.

			The square array A(n,m) (n >= 0, m >= 0) begins:
  1, 1,  1,   1,   1,    1,    1,    1,     1,     1, ...
  1, 2,  4,   6,   8,   10,   12,   14,    16,    18, ...
  1, 4, 16,  36,  64,  100,  144,  196,   256,   324, ...
  1, 6, 36, 114, 264,  510,  876, 1386,  2064,  2934, ...
  1, 8, 64, 264, 768, 1800, 3648, 6664, 11264, 17928, ...
.
The triangle T(n,m) (0 <= m <= n) begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  2,   1;
  [3] 1,  4,   4,   1;
  [4] 1,  6,  16,   6,    1;
  [5] 1,  8,  36,  36,    8,    1;
  [6] 1, 10,  64, 114,   64,   10,   1;
  [7] 1, 12, 100, 264,  264,  100,  12,   1;
  [8] 1, 14, 144, 510,  768,  510, 144,  14,  1;
  [9] 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1;

Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Circular analog of A008288.
Main diagonal: A361743.
Row sums: A361758.
Cf. A142978, A104698.

A := (n, k) -> `if`(n*k=0, 1, 2*n*k*hypergeom([1 - n, 1 - k], [2], 2)):
seq(print(seq(simplify(A(n, k)), k = 0..9)), n=0..4); # Peter Luschny, Mar 23 2023

a[n_Integer?Positive, m_Integer?Positive] := Sum[k Binomial[n, k] Binomial[m, k] 2^k, {k, 1, Min[n,m]}]

from math import comb
def A361745_A(n,m): # compute square array A(n,m)
    return 1 if not(m and n) else sum(comb(n-1,i)*comb(m+i,n) for i in range(max(n-m,0),n))*n<<1 # Chai Wah Wu, Mar 23 2023

A(n,m) = A(m,n).
A(n,m) = Sum_{k=0..min(n,m)} binomial(n,k)*binomial(m,k)*k*2^k for n >= 1.
A(n,m) = n*(D(n,m-1) + D(n-1,m-1)) = n*(D(n,m) - D(n-1,m)) for n,m >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.
G.f.: 2*x*y/(1-x-y-x*y)^2 (valid for n,m > 1).
For n,m >= 1, A(n,m) = 2*n*A142978(n,m).
A(n,m) = 2*n*m*hypergeom([1-n, 1-m], [2], 2) for n,m >= 1. - Peter Luschny, Mar 23 2023

A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]Binomial[n - 1, n - k]2^kx^k, {k, 0, n}]/(2x).

Original entry on oeis.org

1, 2, 2, 3, 12, 4, 4, 36, 48, 8, 5, 80, 240, 160, 16, 6, 150, 800, 1200, 480, 32, 7, 252, 2100, 5600, 5040, 1344, 64, 8, 392, 4704, 19600, 31360, 18816, 3584, 128, 9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256, 10, 810, 17280, 141120, 508032

Offset: 0

Roger L. Bagula, Feb 04 2009

			1;
2, 2;
3, 12, 4;
4, 36, 48, 8;
5, 80, 240, 160, 16;
6, 150, 800, 1200, 480, 32;
7, 252, 2100, 5600, 5040, 1344, 64;
8, 392, 4704, 19600, 31360, 18816, 3584, 128;
9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256;
10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 75-76

A142983, A142978, A047781 (row sums).

Clear[t0, p, x, n, m];
p[x_, n_] = Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
p(x,n)=n Hypergeometric2F1[1 - n, 1 - n, 2, 2 x];
t(n,m)=coefficiemts(p(x,n))
T(n,m) = 2^m*A103371(n,m). - R. J. Mathar, Dec 05 2017

A364429 a(0) = 1, a(n) = (2n^5 + 20n^3 + 23n) 2/15 for n>=1.

Original entry on oeis.org

1, 6, 36, 146, 456, 1182, 2668, 5418, 10128, 17718, 29364, 46530, 71000, 104910, 150780, 211546, 290592, 391782, 519492, 678642, 874728, 1113854, 1402764, 1748874, 2160304, 2645910, 3215316, 3878946, 4648056, 5534766, 6552092, 7713978, 9035328, 10532038, 12221028

Offset: 0

Steven Lu, Jul 24 2023

a(n) is the 6th n-orthoplex (n-dimensional cross-polytope) number.

			a(3) = 146 since the 6th octahedral number is 146; A005900(6) = 146.
a(4) = 456 since the 6th 16-cell number is 456; A014820(5) = 456.

OEIS Wiki, Orthoplex numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005900, A014820, A069038, A069039, A099193, A099195, A099196, A099197, A058331.

Cf. A142978 (column 6 with an initial 1).

Prepend[Table[2/15 (2 x^5 + 20 x^3 + 23 x), {x, 100}], 1]

print([1]+[(2*i**5+20*i**3+23*i)*2//15 for i in range(1,101)])

a(0) = 1, a(n) = (2*n^5 + 20*n^3 + 23*n) * 2/15 for n>=1.

G.f.: (1 + 15*x^2 + 15*x^4 + x^6)/(1 - x)^6. - Stefano Spezia, Jul 24 2023

cp's OEIS Frontend

A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

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A217873 a(n) = 4*n*(n^2 + 2)/3.

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A072410 Number of iterations of the map k -> A000001(k) needed to reach 1 starting at n, or -1 if no such number exists.

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A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

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A361745 Square array of circular Delannoy numbers A(i,j) (i >= 0, j >= 0) read by antidiagonals.

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A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x).

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A364429 a(0) = 1, a(n) = (2*n^5 + 20*n^3 + 23*n) * 2/15 for n>=1.

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Programs

A217873 a(n) = 4n(n^2 + 2)/3.

A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]Binomial[n - 1, n - k]2^kx^k, {k, 0, n}]/(2x).

A364429 a(0) = 1, a(n) = (2n^5 + 20n^3 + 23n) 2/15 for n>=1.