A047926 -id:A047926 - cp's OEIS frontend

A159856 Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.

1, 2, -1, 3, -4, 1, 4, -11, 6, -1, 5, -26, 23, -8, 1, 6, -57, 72, -39, 10, -1, 7, -120, 201, -150, 59, -12, 1, 8, -247, 522, -501, 268, -83, 14, -1, 9, -502, 1291, -1524, 1037, -434, 111, -16, 1, 10, -1013, 3084, -4339, 3598, -1905, 656, -143, 18, -1

Offset: 0

Philippe Deléham, Apr 24 2009

A Riordan array - see the Luzon references.

The second column is A000295 signed. - Michel Marcus, Feb 14 2014

			Triangle begins
  1;
  2,   -1;
  3,   -4,    1;
  4,  -11,    6,   -1;
  5,  -26,   23,   -8,    1;
  6,  -57,   72,  -39,   10,   -1;
  7, -120,  201, -150,   59,  -12,    1;
  ...

Ana Luzón, Iterative Processes Related to Riordan Arrays: The Reciprocation and the Inversion of Power Series, arXiv:0907.2328 [math.CO]; Discrete Math., 310 (2010), 3607-3618.
Ana Luzón and Manuel A. Morón, Riordan matrices in the reciprocation of quadratic polynomials, Linear Algebra Appl. 430 (2009), no. 8-9, 22542270.

Cf. A000295, A047926, A080956, A181690.

With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
+y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)

T(n,k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k,x,0,15),x,n); /* Vladimir Kruchinin, Nov 22 2016 */

From R. J. Mathar, May 31 2009: (Start)
Sum_{k=0..n} T(n,k) = A080956(n).
Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k-1) for k > 0. - Vladimir Kruchinin, Nov 22 2016 [corrected by Werner Schulte, May 09 2024]
G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - Vladimir Kruchinin, Nov 22 2016

a(41) corrected by Georg Fischer, Feb 18 2020

A185055 Number of representations of 5^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c.

Original entry on oeis.org

0, 0, 2, 14, 76, 388, 1950, 9762, 48824, 244136, 1220698, 6103510, 30517572, 152587884, 762939446, 3814697258, 19073486320, 95367431632, 476837158194, 2384185791006, 11920928955068, 59604644775380, 298023223876942, 1490116119384754, 7450580596923816, 37252902984619128

Offset: 0

Zak Seidov, Mar 02 2012

Corresponding formulas for several first primes:
p=3, a(n)=(3*3^n+2*n+1)/4 (A047926)
p=5, a(n)=(5^n-4*n-1)/8 (A185055)
p=7, a(n)=(7^n-1)/6
p=11, a(n)=(3*11^n+10*n-3)/20
p=13, a(n)=(13^n-4*n-1)/8
p=17, a(n)=(17^n-1)/8
p=19, a(n)=(5*19^n+18*n-5)/36
p=23, a(n)=3*(23^n-1)/22
p=29, a(n)=(29^n-4*n-1)/8
p=31, a(n)=2*(31^n-1)/15
p=37, a(n)=(37^n-4*n-1)/8
p=41, a(n)=(41^n-1)/8
p=43, a(n)=(11*43^n+42*n-11)/84
p=47, a(n)=3*(47^n-1)/23.
General formulas for a(n) depend on p mod 8 as follows:
p = 1 mod 8, a(n)=(p^n-1)/8
p = 3 mod 8, a(n)=((p + 1)*p^n + 4*(p - 1)*n - (p + 1))/(8*(p - 1))
p = 5 mod 8, a(n)=(p^n-4*n-1)/8
p = 7 mod 8, a(n)=((p + 1)*(p^n - 1))/(8*(p - 1)).

			a(2)=2 because 25^2 = 9^2+12^2+20^2 = 12^2+15^2+16^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A047926.

a(n) = (5^n-4n-1)/8.
From Chai Wah Wu, Jun 07 2024: (Start)
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2.
G.f.: -2*x^2/((x - 1)^2*(5*x - 1)). (End)
a(n) = 2 * A014827(n-1) for n >= 2. - Alois P. Heinz, Jun 07 2024

A244762 a(n) = (53^n-2n-1)/4.

Original entry on oeis.org

1, 3, 10, 32, 99, 301, 908, 2730, 8197, 24599, 73806, 221428, 664295, 1992897, 5978704, 17936126, 53808393, 161425195, 484275602, 1452826824, 4358480491, 13075441493, 39226324500, 117678973522, 353036920589, 1059110761791, 3177332285398, 9531996856220, 28595990568687, 85787971706089, 257363915118296

Offset: 0

Franklin T. Adams-Watters, Jul 06 2014

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A060816 (first differences).

Cf. A108765, A164039, A047926, A000340.

CoefficientList[Series[(1-2*x+2*x^2)/((1-3*x)*(1-x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2014 *)

a(n+1) = 3*a(n) + n.
G.f.: (1-2*x+2*x^2) / ((1-3*x)*(1-x)^2).
E.g.f.: exp(x)*(5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Aug 28 2023

A103177 a(n) = (7*3^n + 2n + 5)/4.

Original entry on oeis.org

3, 7, 18, 50, 145, 429, 1280, 3832, 11487, 34451, 103342, 310014, 930029, 2790073, 8370204, 25110596, 75331771, 225995295, 677985866, 2033957578, 6101872713, 18305618117, 54916854328, 164750562960, 494251688855, 1482755066539

Offset: 0

Creighton Dement, Mar 17 2005

A floretion-generated sequence relating sequences A007051 and A047926. (a(n)) results from a force transform of the sequence of powers of 3.

Floretion Algebra Multiplication Program, FAMP Code: 4jesleftforseq[A*B] with A = - .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e and B = + 'i + .5j' + .5k' + .5'ij' + .5'ik', 1vesforseq[A*B](n) = 3^(n+1), ForType: 1A.

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A007051, A047926.

Table[(7*3^n+2*n+5)/4,{n,0,25}]  (* Harvey P. Dale, Feb 23 2011 *)

a(n) = 2*A007051(n) + A047926(n).

a(n) = 5*a(n-1)-7*a(n-2)+3*a(n-3). G.f.: (3-2*x)*(1-2*x)/((1-x)^2*(1-3*x)). [Colin Barker, Aug 28 2012]

More terms from Harvey P. Dale, Feb 23 2011

A133341 A007318 * A134312.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 6, 8, 4, 5, 10, 20, 20, 8, 6, 15, 40, 60, 48, 16, 7, 21, 70, 140, 168, 112, 32, 8, 28, 112, 280, 448, 448, 256, 64, 9, 36, 168, 504, 1008, 1344, 1152, 576, 128, 10, 45, 240, 840, 2016, 3360, 3840, 2880, 1280, 256

Offset: 0

Gary W. Adamson, Oct 19 2007

Row sums = A047926: (1, 3, 8, 22, 63, 185, ...).

			First few rows of the triangle:
  1;
  2,   1;
  3,   3,   2;
  4,   6,   8,   4;
  5,  10,  20,  20,   8;
  6,  15,  40,  60,  48,  16;
  7,  21,  70, 140, 168, 112,  32;
  ...

Stijn Vandamme, On the derivatives of the powers of trigonometric and hyperbolic sine and cosine, arXiv:1911.01386 [math.GM], 2019.

Cf. A134312, A047926.

Binomial transform of A134312, as infinite lower triangular matrices.

cp's OEIS Frontend

A159856 Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.

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A185055 Number of representations of 5^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c.

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A244762 a(n) = (5*3^n-2*n-1)/4.

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A103177 a(n) = (7*3^n + 2n + 5)/4.

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A133341 A007318 * A134312.

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A244762 a(n) = (53^n-2n-1)/4.