A002417 -id:A002417 - cp's OEIS frontend

A100431 Bisection of A002417.

8, 80, 336, 960, 2200, 4368, 7840, 13056, 20520, 30800, 44528, 62400, 85176, 113680, 148800, 191488, 242760, 303696, 375440, 459200, 556248, 667920, 795616, 940800, 1105000, 1289808, 1496880, 1727936, 1984760, 2269200, 2583168, 2928640, 3307656, 3722320

Offset: 0

N. J. A. Sloane, Nov 20 2004

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, A002417, A014105, A098077, A108678.

[4*(n+1)^2*(n+2)*(2*n+3)/3: n in [0..60]]; // G. C. Greubel, Apr 09 2023

Table[4*(n+1)^2(n+2)(2n+3)/3, {n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)

[4*(n+1)^2*(n+2)*(2*n+3)/3 for n in range(61)] # G. C. Greubel, Apr 09 2023

a(n) = (4/3)*(2*n^4 + 11*n^3 + 22*n^2 + 19*n + 6). - Ralf Stephan, May 15 2007
G.f.: 8*(1 + 5*x + 2*x^2)/(1 - x)^5. - Ilya Gutkovskiy, Feb 24 2017
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (8/3)*binomial(n+2, 2)*binomial(2*n+3, 2).
a(n) = (8/3)*A000217(n+1)*A014105(n+1).
a(n) = 8*A108678(n).
a(n) = 4*A098077(n+1).
E.g.f.: (4/3)*(6 + 54*x + 69*x^2 + 23*x^3 + 2*x^4)*exp(x). (End)

More terms from Hugo Pfoertner, Nov 26 2004

A100430 Bisection of A002417.

Original entry on oeis.org

1, 30, 175, 588, 1485, 3146, 5915, 10200, 16473, 25270, 37191, 52900, 73125, 98658, 130355, 169136, 215985, 271950, 338143, 415740, 505981, 610170, 729675, 865928, 1020425, 1194726, 1390455, 1609300, 1853013, 2123410, 2422371, 2751840

Offset: 0

N. J. A. Sloane, Nov 20 2004

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. A000447, A002417, A100431.

[(2*n+1)*Binomial(2*n+3,3): n in [0..50]]; // G. C. Greubel, Apr 09 2023

Table[(2*n+1)*Binomial[2*n+3,3], {n,0,50}] (* G. C. Greubel, Apr 09 2023 *)

[(2*n+1)*binomial(2*n+3,3) for n in range(51)] # G. C. Greubel, Apr 09 2023

a(n) = (8*n^4 +28*n^3 +34*n^2 +17*n+3)/3. - Ralf Stephan, May 15 2007
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (2*n+1)*binomial(2*n+3, 3).
a(n) = (2*n+1)*A000447(n+1).
G.f.: (1 + 25*x + 35*x^2 + 3*x^3)/(1-x)^5.
E.g.f.: (1/3)*(3 + 87*x + 174*x^2 + 76*x^3 + 8*x^4)*exp(x). (End)

More terms from Hugo Pfoertner, Nov 26 2004

A281454 Exponential transform of the 4-dimensional figurate numbers (A002417).

Original entry on oeis.org

1, 1, 9, 55, 441, 4316, 46867, 566714, 7550601, 109118728, 1696640501, 28209128344, 498557098921, 9320449092072, 183575505529431, 3796015849264216, 82156098504947473, 1856012774517648896, 43663382492497648777, 1067393396478808265656, 27062739020373087036281, 710410408414549934445376, 19277762831507022675509139

Offset: 0

Ilya Gutkovskiy, Jan 22 2017

nonn

			E.g.f.: A(x) = 1 + x/1! + 9*x^2/2! + 55*x^3/3! + 441*x^4/4! + 4316*x^5/5! + 46867*x^6/6! + ...

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Exponential Transform

Cf. A002417, A281231.

Range[0, 22]! CoefficientList[Series[Exp[Exp[x] x (6 + 18 x + 9 x^2 + x^3)/6], {x, 0, 22}], x]

E.g.f.: exp(exp(x)*x*(6 + 18*x + 9*x^2 + x^3)/6).

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

Original entry on oeis.org

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.

A068239 1/2 the number of colorings of a 3 X 3 square array with n colors.

Original entry on oeis.org

1, 123, 4806, 71410, 583455, 3232341, 13675228, 47502036, 141991245, 377162335, 910842306, 2033854758, 4253012491, 8411348505, 15856955640, 28673921896, 49991146713, 84387303171, 138412872190, 221253017370, 345558093111, 528471784093, 792890261076

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> (79+(-323+(594+(-648+(459+(-216+(66+(-12+n)*n)*n) *n)*n)*n)*n)*n) *n/2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012: (Start)
G.f.: x^2*(1199*x^7 +16567*x^6 +60099*x^5 +71075*x^4 +28765*x^3 +3621*x^2 +113*x+1) / (x-1)^10.
a(n) = (79*n -323*n^2 +594*n^3 -648*n^4 +459*n^5 -216*n^6 +66*n^7 -12*n^8 +n^9) / 2.
(End)

A068305 1/16 the number of colorings of an n X n octagonal array with 16 colors.

Original entry on oeis.org

1, 2730, 1097599230, 65013773510046270, 567344965666217922692546310, 729405912578031916581095228654013174510, 138156586653036397048665899068285784224754055444205830, 3855264256604335001270711977271948484283006596889356139648293174199350

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(5)-a(8) from Alois P. Heinz, May 04 2012

A080851 Square array of pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)

A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 2, 6, 4, 5, 1, 1, 4, 3, 10, 5, 6, 1, 1, 2, 9, 4, 15, 6, 7, 1, 1, 4, 3, 16, 5, 21, 7, 8, 1, 1, 3, 10, 4, 25, 6, 28, 8, 9, 1, 1, 4, 6, 20, 5, 36, 7, 36, 9, 10, 1, 1, 2, 9, 10, 35, 6, 49, 8, 45, 10, 11, 1, 1, 6, 3, 16, 15, 56, 7, 64, 9, 55, 11, 12, 1

Offset: 1

Stefano Spezia, May 19 2020

T(n, k) is called the generalized divisor function (see Beekman).

As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - Gus Wiseman, Aug 04 2022

			From _Gus Wiseman_, Aug 04 2022: (Start)
Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4   9  16  25  36  49  64  81
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
The T(4,5) = 21 chains:
  (1,1,1,1,1)  (4,2,1,1,1)  (4,4,2,2,2)
  (2,1,1,1,1)  (4,2,2,1,1)  (4,4,4,1,1)
  (2,2,1,1,1)  (4,2,2,2,1)  (4,4,4,2,1)
  (2,2,2,1,1)  (4,2,2,2,2)  (4,4,4,2,2)
  (2,2,2,2,1)  (4,4,1,1,1)  (4,4,4,4,1)
  (2,2,2,2,2)  (4,4,2,1,1)  (4,4,4,4,2)
  (4,1,1,1,1)  (4,4,2,2,1)  (4,4,4,4,4)
The T(6,3) = 16 chains:
  (1,1,1)  (3,1,1)  (6,2,1)  (6,6,1)
  (2,1,1)  (3,3,1)  (6,2,2)  (6,6,2)
  (2,2,1)  (3,3,3)  (6,3,1)  (6,6,3)
  (2,2,2)  (6,1,1)  (6,3,3)  (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  3  3  4  1
  1  2  6  4  5  1
  1  4  3 10  5  6  1
  1  2  9  4 15  6  7  1
  1  4  3 16  5 21  7  8  1
  1  3 10  4 25  6 28  8  9  1
  1  4  6 20  5 36  7 36  9 10  1
  1  2  9 10 35  6 49  8 45 10 11  1
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 150 antidiagonals of the array, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
The strict case is A343662 (row sums: A337256).
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291 counts divisors by Omega.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.

T[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; Table[T[n-k,k],{n,1,13},{k,0,n-1}]//Flatten

T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020

T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).

Duplicate term removed by Stefano Spezia, Jun 03 2020

A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

Original entry on oeis.org

1, 1, 3, 1, 8, 6, 1, 15, 30, 10, 1, 24, 90, 80, 15, 1, 35, 210, 350, 175, 21, 1, 48, 420, 1120, 1050, 336, 28, 1, 63, 756, 2940, 4410, 2646, 588, 36, 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45, 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55

Offset: 0

Wolfdieter Lang, Jun 19 2001

Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).

			Triangle starts:
  n\k 0...1.....2......3..... 4.....;
  [0] 1;
  [1] 1,  3;
  [2] 1,  8,    6;
  [3] 1, 15,   30,    10;
  [4] 1, 24,   90,    80,    15;
  [5] 1, 35,  210,   350,   175,    21;
  [6] 1, 48,  420,  1120,  1050,   336,    28;
  [7] 1, 63,  756,  2940,  4410,  2646,   588,    36;
  [8] 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;
  [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Sums include: A001791 (row), (-1)^n*A089849(n+1) (alternating sign row).
Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
Columns: A005563 (k=1), A033487 (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
Cf. A000108, A089849.

A062196:= func;
[A062196(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 21 2025

T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Sep 30 2021

A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
Table[A062196[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 21 2025 *)

def A062196(n,k): return binomial(n,k)*binomial(n+2,k)
print(flatten([[A062196(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 21 2025

T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - Vladimir Kruchinin, Apr 06 2018
From G. C. Greubel, Feb 21 2025: (Start)
T(2*n, n) = (n+1)^2*A000108(n)*A000108(n+1).
T(2*n-1, n) = (4*n^2 - 1)*A000108(n-1)*A000108(n), n >= 1.
T(2*n+1, n) = (1/2)*binomial(n+2,2)*A000108(n+1)*A000108(n+2). (End)

New name by Peter Luschny, Sep 30 2021

A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440

Offset: 0

N. J. A. Sloane, Jan 25 2001

T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012
Conjecture: the matrix inverse is A137452. - R. J. Mathar, Mar 12 2013
The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013

			Triangle begins:
1;
0,  1;
0,  2,   1;
0,  3,   6,    1;
0,  4,  24,   12,    1;
0,  5,  80,   90,   20,   1;
0,  6, 240,  540,  240,  30,  1;
0,  7, 672, 2835, 2240, 525, 42,  1;
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983).
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96, [_Tom Copeland_, Dec 20 2018].
Eric Weisstein's World of Mathematics, Abel Polynomial
Eric Weisstein's World of Mathematics, Idempotent Number
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.
Cf. A061356, A202017, A137452 (inverse array), A264428.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n, k) *k^(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 05 2012

nn=10;f[list_]:=Select[list,#>0&];Prepend[Map[Prepend[#,0]&,Rest[Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x Exp[x]],{x,0,nn}],{x,y}]]]],{1}]//Grid  (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

# uses[bell_transform from A264428]
def A059297_row(n):
    nat = [k for k in (1..n)]
    return bell_transform(n, nat)
[A059297_row(n)  for n in range(8)] # Peter Luschny, Dec 20 2015

E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic, Nov 18 2003
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011
The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011
Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015
The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - Peter Luschny, Dec 20 2015

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