A163626 -id:A163626 - cp's OEIS frontend

A228912 a(n) = 10^n - 99^n + 368^n - 847^n + 1266^n - 1265^n + 844^n - 363^n + 92^n - 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 19958400, 618710400, 14270256000, 273158645760, 4595022432000, 70309810771200, 1000944296352000, 13467262000832640, 173201547619900800, 2147373231974006400, 25832386565857872000, 303056981918271947520, 3481253462769108364800

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the tenth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y = 1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Tenth column of results of A163626.
Essentially 362880*A049435.
Cf. A228910 (with more crossrefs), A228911.

Table[9!*StirlingS2[n+1, 10], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[362880*x^9 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)

a(n)=10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1

G.f.: 362880*x^9 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..10} (-1)^(10-k)*binomial(10-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A228913 a(n) = 11^n-1010^n+459^n-1208^n+2107^n-2526^n+2105^n-1204^n+453^n-10*2^n+1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 239500800, 8821612800, 239740300800, 5368729766400, 105006251750400, 1858166876966400, 30449278610150400, 469614684719980800, 6897777008118796800, 97349279409046828800, 1329165939158093836800, 17651395149921751680000

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the eleventh column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (66, -1925, 32670, -357423, 2637558, -13339535, 45995730, -105258076, 150917976, -120543840, 39916800).

Eleventh column of results of A163626.

Cf. A228910 (with more cf.s), A228911, A228912.

Table[10!*StirlingS2[n+1, 11], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[-3628800*x^10 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)*(11*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
LinearRecurrence[{66,-1925,32670,-357423,2637558,-13339535,45995730,-105258076,150917976,-120543840,39916800},{0,0,0,0,0,0,0,0,0,0,3628800},30] (* Harvey P. Dale, Mar 20 2017 *)

a(n)=11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1

G.f.: -3628800*x^10 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)*(11*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..11} (-1)^(11-k)*binomial(11-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A051782 Apply the "Stirling-Bernoulli transform" to Catalan numbers.

Original entry on oeis.org

1, 0, 2, -12, 122, -1620, 26882, -536172, 12506762, -334261380, 10075002962, -338180323932, 12512502202202, -505992958647540, 22204726014875042, -1050993549782729292, 53373431773793542442, -2894886293042487680100, 167021024758368026331122

Offset: 0

N. J. A. Sloane, Dec 09 1999

sign

The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ... to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has e.g.f. exp(x)*B(1-exp(x)). More explicitly, c_n = Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1,m+1)*b_m.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000108, A163626.

a:= n-> add((-1)^k *k! *Stirling2(n+1, k+1)*binomial(2*k, k)/
        (k+1), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2013

a[n_] := Sum[(-1)^k k! StirlingS2[n+1, k+1] CatalanNumber[k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Apr 06 2016 *)

a(n) = Sum_{k = 0..n} A163626(n,k)*A000108(k). - Philippe Deléham, May 25 2015

A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

Original entry on oeis.org

-1, 2, -2, -6, 9, -3, 24, -44, 24, -4, -120, 250, -175, 50, -5, 720, -1644, 1350, -510, 90, -6, -5040, 12348, -11368, 5145, -1225, 147, -7, 40320, -104544, 105056, -54152, 15680, -2576, 224, -8, -362880, 986256, -1063116, 605556, -202041, 40824, -4914, 324, -9, 3628800, -10265760, 11727000, -7236800

Offset: 1

Vladeta Jovovic, Jun 05 2004

			Triangle starts:
[n\k    1        2       3      4      5      6     7  8]
[1]    -1;
[2]     2,      -2;
[3]    -6,       9,     -3;
[4]    24,     -44,     24,     -4;
[5]  -120,     250,   -175,     50,    -5;
[6]   720,   -1644,   1350,   -510,    90,    -6;
[7] -5040,   12348, -11368,   5145, -1225,   147,   -7;
[8] 40320, -104544, 105056, -54152, 15680, -2576,  224,  -8;

Cf. A019538, A028246, A163626.

Cf. A000142, A052881.

T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1);
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, May 26 2020

Table[StirlingS1[n+1,k]-StirlingS1[n,k-1],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 25 2024 *)

E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]
The matrix inverse of the Worpitzky triangle. More precisely:
T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - Peter Luschny, May 26 2020

Offset of k shifted and edited by Peter Luschny, May 26 2020

A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 7, 12, 6, 0, 1, 15, 50, 60, 24, 0, 1, 31, 180, 390, 360, 120, 0, 1, 63, 602, 2100, 3360, 2520, 720, 0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 0, 1, 511, 18660

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Row n gives the coefficients which express the sums of the n-th powers of the integers as a linear combination of binomial coefficients, thus:
     Sum_{k=1..r} k^n = A103438(n+r,r) = Sum_{k=0..n} T(n+1,k) * C(r,k),
  where, by convention, C(r,k) = 0 whenever r < k. - Robert B Fowler, Jan 16 2023

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    2;
  0, 1,   7,   12,     6;
  0, 1,  15,   50,    60,    24;
  0, 1,  31,  180,   390,   360,   120;
  0, 1,  63,  602,  2100,  3360,  2520,   720;
  0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040;
  ...

Peter Luschny, Row 0 to 44, recomputed Jan 19 2019

Column k = 0 is A000007.
Cf. A028246, A163626, A000629 (row sums).
Cf. A103438, A007318 (binomial coefficients).

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A142071row := proc(n) 1 - log(1 - x*(exp(z) - 1)):
series(%, z, 12): CL(n!*coeff(%, z, n), x) end:
for n from 0 by 1 to 7 do A142071row(n) od;
# Alternative:
A142071Row := proc(n) if n=0 then [1] else
CL(convert(series(polylog(-n+1, z/(1+z)), z, n*2), polynom), z) fi end:
seq(A142071Row(n), n=0..6); # Peter Luschny, Sep 06 2018

T[n_, k_] := If[k==0, Floor[1/(n + 1)], (k - 1)!*StirlingS2[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 06 2024 *)

Row n gives the coefficients of the polynomial defined by p(x, 0) = 1 and for n > 0 p(x, n) = Sum_{k >= 0} k^(n-1)*(x/(1 + x))^k = PolyLog(-n+1, x/(1+x)).

T(n, k) = (k - 1)! * Stirling2(n, k) for k > 0. - Detlef Meya, Jan 06 2024

Edited, T(0,0) = 1 prepended and new name by Peter Luschny, Sep 06 2018

A260196 1, -3, followed by -1's.

Original entry on oeis.org

1, -3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Paul Curtz, Jul 19 2015

1/(n+1) is the inverse Akiyama-Tanigawa transform of A164555(n)/A027642(n).
For more on the Akiyama-Tanigawa transform, see Links (correction: page 7 read here A164555 instead of A027641) and A177427.
Here:
   1,  -3, -1, -1, -1, -1, ...
   4,  -4,  0,  0,  0,  0, ...
   8,  -8,  0,  0,  0,  0, ...
  16, -16,  0,  0,  0,  0, ...
  etc.
Other process, using signed A130534(n), different of A008275(n):
1,                            1/1,                                      1,
1, 4,                      (  1,     -1)/1,                            -3,
1, 4, 8,                   (  2,     -3,   1)/2,                       -1,
1, 4, 8, 16,          *    (  6,    -11,   6,   -1)/6,             =   -1,
1, 4, 8, 16, 32,           ( 24,    -50,  35,  -10,  1)/24,            -1,
1, 4, 8, 16, 32, 64,       (120,   -274, 225,  -85, 15, -1)/120,       -1,
etc.                       etc.                                        etc.
Via the modified Stirling numbers of the first kind, the second triangle, Iw(n), is the inverse of Worpitzky transform A163626(n).
a(n) is the third sequence of a family beginning with
1,  1,  1,  1,  1,  1,  1,  1, ...              = A000012(n)
1,  0,  0,  0,  0,  0,  0,  0,  0, ...          = A000007(n)
1, -3, -1, -1, -1, -1, -1, -1, -1, -1, ... .
A000012(n) is the inverse Akiyama-Tanigawa transform of A000007(n), with or without its second term.
A000007(n) is the inverse Akiyama-Tanigawa transform of A000012(n), with or without its second term.
a(n) is the inverse Akiyama-Tanigawa transform of 2^n omitting the second term i.e. 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, Journal of Integer Sequences, 3(2000), article 00.2.9
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000012, A008275, A027642, A130534, A151821, A163626, A164555, A177427.

first(m)=vector(m,i,i--;if(i>1,-1,if(i==0,1,if(i==1,-3)))) \\ Anders Hellström, Aug 28 2015

Vec(-(2*x^2-4*x+1)/(x-1) + O(x^100)) \\ Colin Barker, Sep 11 2015

Inverse Akiyama-Tanigawa transform of A151821(n).
From Colin Barker, Sep 11 2015: (Start)
a(n) = -1 for n>1.
a(n) = a(n-1) for n>2.
G.f.: -(2*x^2-4*x+1) / (x-1).
(End)

A344920 The Worpitzky transform of the squares.

Original entry on oeis.org

0, -1, 5, -13, 29, -61, 125, -253, 509, -1021, 2045, -4093, 8189, -16381, 32765, -65533, 131069, -262141, 524285, -1048573, 2097149, -4194301, 8388605, -16777213, 33554429, -67108861, 134217725, -268435453, 536870909, -1073741821, 2147483645, -4294967293

Offset: 0

Peter Luschny, Jun 24 2021

The Worpitzky transform maps a sequence A to a sequence B, where B(n) = Sum_{k=0..n} A163626(n, k)*A(k). (If A(n) = 1/(n + 1) then B(n) are the Bernoulli numbers (with B(1) = 1/2.))

Also row 2 in A371761. Can be generated by the signed Akiyama-Tanigawa algorithm for powers (see the Python script). - Peter Luschny, Apr 12 2024

Index entries for linear recurrences with constant coefficients, signature (-3,-2).

Up to shift and sign: even bisection A267921, odd bisection A141725.

Cf. A163626, A036563, A371761.

gf := (exp(x) - 1)*(exp(x) - 2)*exp(-2*x): ser := series(gf, x, 36):
seq(n!*coeff(ser, x, n), n = 0..31);

W[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];
WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];
WT[#^2 &, 32] (* The Worpitzky transform applied to the squares. *)

# Using the Akiyama-Tanigawa algorithm for powers from A371761.
print([(-1)**n * v for (n, v) in enumerate(ATPowList(2, 32))])
# Peter Luschny, Apr 12 2024

a(n) = n! * [x^n] (exp(x) - 1)*(exp(x) - 2)*exp(-2*x).
a(n) = (-1)^(n + 1)*(3 - 2^(n + 1)) for n >= 1. - Hugo Pfoertner, Jun 24 2021
a(n) = [x^n] x*(2*x - 1)/(2*x^2 + 3*x + 1). - Stefano Spezia, Jun 24 2021

A258369 Stirling-Bernoulli transform of A027656.

Original entry on oeis.org

1, 1, 5, 25, 173, 1441, 14165, 160105, 2044733, 29105521, 456781925, 7834208185, 145760370893, 2923764916801, 62891469229685, 1444055265984265, 35250519098274653, 911569049328779281, 24893164161460525445, 715822742720760256345, 21620050147748210572013

Offset: 0

Philippe Deléham, May 28 2015

Also called Akiyama-Tanigawa transform of A027656.

			a(0) = 1*1 = 1.
a(1) = 1*1 = 1.
a(2) = 1*1 + 2*2 = 5.
a(3) = 1*1 + 12*2 = 25.
a(4) = 1*1 + 50*2 + 24*3 = 173.

Cf. A027656, A163626, A249163.

a(n) = Sum_{k = 0..n} A163626(n,k)*A027656(k).
a(n) = Sum_{k>=0} A249163(n,k) * (k+1).
E.g.f.: 1/(exp(x)*(2 - exp(x))^2).
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018

A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.

The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).

			[0] [      1]
[1] [     -1,         1]
[2] [      5,       -11,        6]
[3] [    -61,       211,     -240,        90]
[4] [   1385,     -6551,    11466,     -8820,     2520]
[5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.

Row sums are A000007, alternating row sums are A210657.
Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);

Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0,k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)

def EW(m, n):
    @cached_function
    def S(m, n):
        R. = ZZ[]
        if n == 0: return R(1)
        return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
    s = S(m, n).list()
    c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
        sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
    return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])

Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:

T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).

A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 19, -39, 20, -1513, 4705, -4872, 1680, 315523, -1314807, 2052644, -1422960, 369600, -136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000, 105261234643, -661231439271, 1729495989332, -2410936679424, 1889230062720, -789044256000, 137225088000

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See A318259 and the cross-references for some other members.

			[0] [         1]
[1] [        -1,         1]
[2] [        19,       -39,          20]
[3] [     -1513,      4705,       -4872,       1680]
[4] [    315523,  -1314807,     2052644,   -1422960,     369600]
[5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]

Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.

Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).

# uses[EW from A318259]
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))

Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1

and S(n,k) = [x^k]P(3,n), then T(n,k) = Sum_{j=0..k}((-1)^(k-j)*binomial(n-j, n-k)* Sum_{i=0..n}((-1)^i*S(n,i)*binomial(n-i,j))).

cp's OEIS Frontend

A228912 a(n) = 10^n - 9*9^n + 36*8^n - 84*7^n + 126*6^n - 126*5^n + 84*4^n - 36*3^n + 9*2^n - 1.

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A228913 a(n) = 11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1.

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Mathematica

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A051782 Apply the "Stirling-Bernoulli transform" to Catalan numbers.

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Maple

Mathematica

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A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

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Maple

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A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

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Maple

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A260196 1, -3, followed by -1's.

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PARI

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A344920 The Worpitzky transform of the squares.

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Maple

A228912 a(n) = 10^n - 99^n + 368^n - 847^n + 1266^n - 1265^n + 844^n - 363^n + 92^n - 1.

A228913 a(n) = 11^n-1010^n+459^n-1208^n+2107^n-2526^n+2105^n-1204^n+453^n-10*2^n+1.