A002697 -id:A002697 - cp's OEIS frontend

A041001 Convolution of A000108(n+1), n >= 0, (Catalan numbers) with A038845 (3-fold convolution of powers of 4).

1, 14, 125, 906, 5810, 34364, 191901, 1026610, 5312230, 26767940, 131990066, 639210404, 3048892740, 14354652152, 66828135005, 308078809794, 1408022619806, 6385966846580, 28765327498278, 128777533131500

Offset: 0

Wolfdieter Lang

Also convolution of A038836 with A000984 (central binomial coefficients); also convolution of A001791(n+1), n >= 0, with A002802; also convolution of A008549(n+1), n >= 0, with A002697; also convolution of A029760 with A002457; also convolution of A038806(n+1), n >= 0, with A000302 (powers of 4).

a(n) = (n+3)*(3*(n+6)*2^(2*n+3)-(n+4)*binomial(2*n+7, n+3))/12; G.f. (c(x)^2)/(1-4*x)^3, where c(x) = g.f. for Catalan numbers.

A143376 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1 <= k <= n).

Original entry on oeis.org

1, 4, 2, 12, 12, 4, 32, 48, 32, 8, 80, 160, 160, 80, 16, 192, 480, 640, 480, 192, 32, 448, 1344, 2240, 2240, 1344, 448, 64, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256, 5120, 23040, 61440, 107520

Offset: 1

Emeric Deutsch, Sep 05 2008

Sum of entries in row n = 2^(n-1)*(2^n-1) = A006516.
The entries in row n are the coefficients of the Wiener polynomial of the cube Q_n.
Sum_{k=1..n} k*T(n,k) = n*4^(n-1) = A002697(n) = the Wiener index of the cube Q_n.
Triangle T(n,k), 1 <= k <= n, read by rows given by [1,1,0,0,0,0,0,...]DELTA[1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938; subtriangle of triangle A055372. - Philippe Deléham, Oct 14 2008

			T(2,1)=4, T(2,2)=2 because in Q_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2.
Triangle starts:
   1;
   4,   2;
  12,  12,   4;
  32,  48,  32,   8;
  80, 160, 160,  80,  16;

Indranil Ghosh, Rows 1..125, flattened
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A002697, A006516.

T:=proc(n,k) options operator, arrow: 2^(n-1)*binomial(n,k) end proc: for n to 10 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

nn = 8; A[u_, z_] := (z + u z)/(1 - (z + u z));
Drop[Map[Select[#, # > 0 &] &, Map[Drop[#, 1] &,CoefficientList[Series[1/(1 - A[u, z]), {z, 0, nn}], {z, u}]]],1] // Grid (* Geoffrey Critzer, Mar 04 2017 *)
Flatten[Table[2^(n-1) Binomial[n, k], {n, 10},{k,n}]] (* Indranil Ghosh, Mar 06 2017 *)

tabl(nn) = {for (n=1, nn, for(k=1, n, print1(2^(n-1) * binomial(n, k),", ");); print();); };
tabl(10); \\ Indranil Ghosh, Mar 06 2017

import math
f=math.factorial
def C(n,r): return f(n) / f(r) / f(n-r)
i=1
for n in range(1,126):
    for k in range(1,n+1):
        print(str(i)+" "+str(2**(n-1)*C(n,k)))
        i+=1 # Indranil Ghosh, Mar 06 2017

T(n,k) = 2^(n-1)*binomial(n,k).
G.f.: G(q,z) = qz/((1-2z)(1-2z-2zq)).
T(n,k) = A055372(n,k). - Philippe Deléham, Oct 14 2008

Typo corrected by Philippe Deléham, Jan 05 2009

A038697 Convolution of A000917 with A000984 (central binomial coefficients).

Original entry on oeis.org

3, 26, 163, 894, 4558, 22196, 104739, 483062, 2189530, 9789900, 43295118, 189749676, 825364668, 3567219688, 15332925731, 65591312550, 279415474594, 1185903736412, 5016725589402, 21159849864964, 89012979703940

Offset: 0

Wolfdieter Lang

Also convolution of A007054 (Super ballot numbers) with A002697;

Robert Israel, Table of n, a(n) for n = 0..1652

Cf. A007054, A038665, A038679, A000917, A000108, A000984, A002697.

seq(n*4^(n+1)+binomial(2*n+3,n+1),n=0..30); # Robert Israel, May 22 2019

a(n) = n*4^(n+1)+binomial(2*n+3, n+1).
G.f.: c(x)*(4-c(x))/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108.
(160+64*n)*a(n) - (160+48*n)*a(n+1) + (50+12*n)*a(n+2) - (5+n)*a(n+3)=0. - Robert Israel, May 22 2019

A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

Original entry on oeis.org

1, 16, 159, 1260, 8722, 55152, 326811, 1844084, 10015566, 52754624, 270976342, 1362986520, 6734927460, 32775704608, 157408497171, 747269225028, 3511471892470, 16351481223840, 75525932249922, 346305571781224

Offset: 0

Wolfdieter Lang

Also convolution of A001791(n+1), n >= 0, with A038845; also convolution of A008549(n+1), n >= 0, with A002802; also convolution of A029760 with A002697; also convolution of A038806(n+1), n >= 0, with A002457; also convolution of A038836 with A000302 (powers of 4); also convolution of A041001 with A000984 (central binomial coefficients).

a(n)=binomial(n+7, 3)*binomial(2*(n+4), n+2)/20 - (n+4)*(n+3)*4^(n+1); G.f. (c(x)^2)/(1-4*x)^(7/2), where c(x) = g.f. for Catalan numbers.

A087449 a(n) = n * 4^(n-1) + (2*4^n + 1) / 3.

Original entry on oeis.org

1, 4, 19, 91, 427, 1963, 8875, 39595, 174763, 764587, 3320491, 14330539, 61516459, 262843051, 1118481067, 4742359723, 20043180715, 84467690155, 355050629803, 1488921995947, 6230565890731, 26021775190699, 108485147273899

Offset: 0

Paul Barry, Sep 05 2003

Binomial transform of A064017.

Index entries for linear recurrences with constant coefficients, signature (9,-24,16).

Cf. A002697, A007583, A064017.

LinearRecurrence[{9,-24,16},{1,4,19},30] (* Harvey P. Dale, Apr 15 2018 *)

a(n) = my(p4 = 1<<(2*n)); n * p4 / 4 + (2*p4 + 1) / 3 \\ David A. Corneth, Apr 15 2018

G.f.: (1-5x+7x^2)/((1-x)(1-4x)^2).

a(n) = A002697(n) + A007583(n).

Name clarified by David A. Corneth, Apr 15 2018

A106691 Expansion of g.f. (1+x-2x^2+x^3+x^4)/((1-x)^2(1+x)^2(1+2x)^2).

Original entry on oeis.org

1, -3, 8, -17, 36, -71, 140, -269, 516, -979, 1852, -3481, 6516, -12127, 22444, -41253, 75236, -135915, 242716, -427185, 737876, -1242743, 2019468, -3106877, 4349636, -4971011, 2485500, 9942071, -49710284, 159072881, -437450388, 1113510059, -2704238684, 6362914533, -14634703396

Offset: 0

Creighton Dement, May 13 2005

Floretion Algebra Multiplication Program, FAMP Code: 2jbasekrokseq[ - .25'i - .25i' + 'ii' + .25'jk' + .25'kj'], RokType: Y[sqa.Findk()] = Y[sqa.Findk()] - p (internal program code)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-2,8,7,-4,-4).

Cf. A002697.

[(1/54)*(3*n +4 -27*(-1)^n*(n+4) +(-2)^(n+1)*(3*n-79)): n in [0..40]]; // G. C. Greubel, Sep 09 2021

CoefficientList[Series[(1+x-2x^2+x^3+x^4)/((1-x)^2(1+x)^2(1+2x)^2),{x,0,40}],x] (* or *) LinearRecurrence[{-4,-2,8,7,-4,-4},{1,-3,8,-17,36,-71},40] (* Harvey P. Dale, Dec 21 2015 *)

[(1/54)*(3*n +4 -27*(-1)^n*(n+4) +(-2)^(n+1)*(3*n-79)) for n in (0..40)] # G. C. Greubel, Sep 09 2021

From G. C. Greubel, Sep 09 2021: (Start)
a(n) = (1/54)*(3*n +4 -27*(-1)^n*(n+4) +(-2)^(n+1)*(3*n-79)).
E.g.f.: (1/54)*((4 +3*x)*exp(x) -27*(4 -x)*exp(-x) + 2*(79 +6*x)*exp(-2*x)). (End)

A172978 a(n) = binomial(n+10, 10)*4^n.

Original entry on oeis.org

1, 44, 1056, 18304, 256256, 3075072, 32800768, 318636032, 2867724288, 24216338432, 193730707456, 1479398129664, 10848919617536, 76776969601024, 526470648692736, 3509804324618240, 22813728110018560, 144934272698941440, 901813252348968960, 5505807224867389440

Offset: 0

Zerinvary Lajos, Feb 06 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..157
Index entries for linear recurrences with constant coefficients, signature (44,-880,10560,-84480,473088,-1892352,5406720,-10813440,14417920,-11534336,4194304).

Cf. A002697, A038845, A038846, A040075, A045543, A054337, A054338, A054339, A054340

[Binomial(n+10, 10)*4^n: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Table[Binomial[n + 10, 10]*4^n, {n, 0, 20}]

From Amiram Eldar, Mar 27 2022: (Start)
G.f.: 1/(1 - 4*x)^11.
Sum_{n>=0} 1/a(n) = 14269429/63 - 787320*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 78125000*log(5/4) - 1098284605/63. (End)

A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 4, 16, 1, 64, 8, 256, 48, 1, 1024, 256, 12, 4096, 1280, 96, 1, 16384, 6144, 640, 16, 65536, 28672, 3840, 160, 1, 262144, 131072, 21504, 1280, 20, 1048576, 589824, 114688, 8960, 240, 1, 4194304, 2621440, 589824, 57344, 2240, 24, 16777216, 11534336, 2949120, 344064, 17920, 336, 1

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.

			Triangle begins:
         1;
         4;
        16,        1;
        64,        8;
       256,       48,        1;
      1024,      256,       12;
      4096,     1280,       96,       1;
     16384,     6144,      640,      16;
     65536,    28672,     3840,     160,      1;
    262144,   131072,    21504,    1280,     20;
   1048576,   589824,   114688,    8960,    240,    1;
   4194304,  2621440,   589824,   57344,   2240,   24;
  16777216, 11534336,  2949120,  344064,  17920,  336,  1;
  67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.

Shara Lalo, Left justified triangle
Shara Lalo, Skew diagonals in triangle A013611

Row sums give A001076.
Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
Cf. A013611.
Cf. A098317.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

G.f.: 1 / (1 - 4*t*x - t^2).

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

Original entry on oeis.org

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

Offset: 0

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).

			The triangle T begins:
n\k    0       1        2         3         4          5          6         7
0:     1
1:    -3       4
2:     9     -32       25
3:   -27     192     -375       216
4:    81   -1024     3750     -5184      2401
5:  -243    5120   -31250     77760    -84035      32768
6:   729  -24576   234375   -933120   1764735   -1572864     531441
7: -2187  114688 -1640625   9797760 -28824005   44040192  -33480783  10000000
...
n = 8:  6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n,k](k+3)^n,{k,0,n}];Array[A362353row,10,0] (* Paolo Xausa, Jul 30 2023 *)

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

a(41)-a(44) from Paolo Xausa, Jul 31 2023

A368043 Triangle read by rows: T(n, k) = 2^(n + k).

Original entry on oeis.org

1, 2, 4, 4, 8, 16, 8, 16, 32, 64, 16, 32, 64, 128, 256, 32, 64, 128, 256, 512, 1024, 64, 128, 256, 512, 1024, 2048, 4096, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144

Offset: 0

Peter Luschny, Dec 09 2023

			[0]  [  1]
[1]  [  2,   4]
[2]  [  4,   8,  16]
[3]  [  8,  16,  32,    64]
[4]  [ 16,  32,  64,   128,  256]
[5]  [ 32,  64,  128,  256,  512, 1024]
[6]  [ 64, 128,  256,  512, 1024, 2048,  4096]
[7]  [128, 256,  512, 1024, 2048, 4096,  8192, 16384]
[8]  [256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536]

Cf. A000079 (T(n,0)), A004171 (T(n,n-1)), A000302 (T(n,n)), A171476 (row sums), A003683 (alternating row sums), A134353 (antidiagonal sums), A001018 (T(2n, n)), A094014 (T(n, n/2)), A002697.

Array[2^Range[#,2#]&,10,0] (* Paolo Xausa, Dec 09 2023 *)

from functools import cache
@cache
def T_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = T_row(n - 1) + [0]
    for k in range(n): row[k] *= 2
    row[n] = row[n - 1] * 2
    return row
for n in range(11): print(T_row(n))

G.f.: 1/((1 - 2*x)*(1 - 4*x*y)). - Stefano Spezia, Dec 09 2023

cp's OEIS Frontend

A041001 Convolution of A000108(n+1), n >= 0, (Catalan numbers) with A038845 (3-fold convolution of powers of 4).

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A143376 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1 <= k <= n).

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A038697 Convolution of A000917 with A000984 (central binomial coefficients).

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A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

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A087449 a(n) = n * 4^(n-1) + (2*4^n + 1) / 3.

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A106691 Expansion of g.f. (1+x-2*x^2+x^3+x^4)/((1-x)^2*(1+x)^2*(1+2*x)^2).

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A172978 a(n) = binomial(n+10, 10)*4^n.

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A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A106691 Expansion of g.f. (1+x-2x^2+x^3+x^4)/((1-x)^2(1+x)^2(1+2x)^2).

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.