A059304 -id:A059304 - cp's OEIS frontend

A261357 Pyramid of coefficients in expansion of (1 + 2x + 2y)^n.

1, 1, 2, 2, 1, 4, 4, 4, 8, 4, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), two kinds of steps (1,1,0) and two kinds of steps (1,1,1).
The sum of the numbers in each slice of the pyramid is 5^n.
The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each antidiagonal of the j-th triangle of the n-th slice of A261358. - Dimitri Boscainos, Aug 21 2015

			Here is the fourth (n=3) slice of the pyramid:
        1
      6   6
   12  24  12
  8  24  24   8

Alois P. Heinz, Rows n = 0..38, flattened

Cf. A000079, A046816, A059304, A261356, A261358.

p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + 2*p[i - 1, j, k - 1] + 2*p[i - 1, j - 1, k - 1]]];
Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Mar 17 2025, after Alois P. Heinz *)

tabf(nn) = {for (n=0, nn, for (j=0, n, for (k=0, j, print1(2^j*binomial(n,j)*binomial(j,k), ", ")); print();); print(););} \\ Michel Marcus, Oct 07 2015

T(i+1,j,k) = 2*T(i,j-1,k-1)+ 2*T(i,j-1,k) + T(i,j,k); T(i,j,-1) = 0, ...; T(0,0,0) = 1.

T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870

Offset: 0

Peter Luschny, Aug 12 2022

The counterpart using the falling factorial is Leibniz's Harmonic Triangle A003506.

			Triangle T(n, k) begins:
[0]     1;
[1]     2,      2;
[2]     6,     12,      6;
[3]    20,     60,     60,     20;
[4]    70,    280,    420,    280,     70;
[5]   252,   1260,   2520,   2520,   1260,    252;
[6]   924,   5544,  13860,  18480,  13860,   5544,    924;
[7]  3432,  24024,  72072, 120120, 120120,  72072,  24024,   3432;
[8] 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870;

cf. A000984, A059304 (row sums, see also A343842), A265609 (rising factorial).

Cf. A003506, A173018 (Eulerian numbers), A000108, A000897 (central terms).

A356546 := (n, k) -> pochhammer(n+1, n)/(k!*(n-k)!):
for n from 0 to 8 do seq(A356546(n, k), k=0..n) od;

T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* Michael Somos, Aug 18 2022 *)

{T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* Michael Somos, Aug 18 2022 */

def A356546(n, k):
    return rising_factorial(n+1,n) // (factorial(k) * factorial(n-k))
for n in range(9): print([A356546(n, k) for k in range(n+1)])

Bernoulli(n) / Catalan(n) = Sum_{k=0..n} (-1)^k*A173018(n, k) / T(n, k), (with Bernoulli(1) = 1/2).

G.f.: 1/sqrt(1 - 4*x*(y + 1)). - Vladimir Kruchinin, Feb 15 2023

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

Original entry on oeis.org

2, 42, 1376, 60934, 3377252, 224036904, 17282039280, 1519096411230, 149867251224092, 16398595767212452, 1971137737765484444, 258215735255164847944, 36617351885600586385222, 5588967440618883091216208, 913592455995572681826313856, 159241707066923571547572653630

Offset: 1

Paul D. Hanna, Feb 11 2023

nonn

Related sequence: A000984(n) = binomial(2*n,n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for n >= 1.

			L.g.f.: A(x) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + 1519096411230*x^8/8 + ...
a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
Exponentiation yields the g.f. of A360239:
exp(A(x)) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + ... + A360239(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A360239, A360348, A266526, A000984, A059304, A098658.

{a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
for(n=0,20,print1(a(n),", "))

a(n) ~ (1 - exp(-1)/4) * 2^(2*n) * n^(n + 1/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 12 2023

A386843 a(n) = Sum_{k=0..n} binomial(2n+2,k) binomial(2*n-k,n-k).

Original entry on oeis.org

1, 6, 39, 268, 1905, 13842, 102123, 761880, 5732325, 43417630, 330620895, 2528772132, 19412942809, 149497184298, 1154365194195, 8934458916912, 69291946278861, 538372925816886, 4189702003359687, 32651982699233340, 254800541773725633, 1990683254889381954

Offset: 0

Seiichi Manyama, Aug 05 2025

nonn

Cf. A386844, A386845.
Cf. A059304, A178792.
Cf. A116881.

a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k, n-k));

a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+2,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k,k).

A386918 a(n) = 2^n * binomial(4*n,n).

Original entry on oeis.org

1, 8, 112, 1760, 29120, 496128, 8614144, 151557120, 2692684800, 48201359360, 868004380672, 15706806542336, 285362317180928, 5202031080243200, 95104728494899200, 1743063914667048960, 32016101348447354880, 589188508080622534656, 10861173739509105295360

Offset: 0

Seiichi Manyama, Aug 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A066381, A386919, A386920.

Cf. A005810, A059304, A228484, A378804.

[2^n * Binomial(4*n,n): n in [0..26]]; // Vincenzo Librandi, Aug 11 2025

Table[2^n*Binomial[4*n,n],{n,0,30}] (* Vincenzo Librandi, Aug 11 2025 *)

PARI
```
a(n) = 2^n*binomial(4*n, n);
```

a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(4*n)/(1-x)^(3*n+1).
a(n) = [x^n] 1/(1-2*x)^(3*n+1).
a(n) = [x^n] (1+2*x)^(4*n).

A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.

Original entry on oeis.org

1, 6, 4, 42, 60, 24, 308, 688, 560, 160, 2310, 7080, 8760, 5040, 1120, 17556, 68712, 114576, 99456, 44352, 8064, 134596, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1038312, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720

Offset: 0

R. J. Mathar, Mar 17 2007

The integral N(a;n) = Integral_{x=0..infinity} 1/(x^4 + 2*a*x^2 + 1)^(n+1) has a polynomial representation P_n(a) = 2^(n + 3/2) * (a+1)^(n + 1/2) * N(a;n) / Pi (known as the Boros-Moll polynomial). The table contains the coefficients T(n,l) of P_n(a) = 2^(-2*n)*Sum_{l=0..n} T(n,l)*a^l in row n and column l (with n >= 0 and 0 <= l <= n).

			The table T(n,l) (with rows n >= 0 and columns l = 0..n) starts:
      1;
      6,     4;
     42,    60,     24;
    308,   688,    560,   160;
   2310,  7080,   8760,  5040,  1120;
  17556, 68712, 114576, 99456, 44352, 8064;
  ...
For n = 2, N(a;2) = Integral_{x=0..oo} dx/(x^4 + 2*a*x + 1)^3 = 2^(-2*2)*(Sum_{l=0..2} T(2,l)*a^l) * Pi/(2^(2 + 3/2) * (a + 1)^(2 + 1/2) = (42 + 60*a + 24*a^2) * Pi/(32 * (2*(a+1))^(5/2)) for a > -1. - _Petros Hadjicostas_, May 25 2020

Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Victor H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Victor H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin., no 2 (2007), #4.

Cf. A002458 (row sums), A004982 (column l=0), A059304 (main diagonal), A067001 (rows reversed), A223549, A223550, A334907.

A126936 := proc(m, l)
    add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m):
end:
seq(seq(A126936(m,l), l=0..m), m=0..12); # R. J. Mathar, May 25 2020

t[m_, l_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, l, m}]; Table[t[m, l], {m, 0, 11}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Maple, adapted May 2020 *)

From Petros Hadjicostas, May 25 2020: (Start)
T(n,l) = A067001(n, n-l) = 2^(2*n) * A223549(n,l)/A223550(n,l).
Sum_{l=0..n} T(n,l) = A002458(n) = A334907(n)*2^n/n!.
Bivariate o.g.f.: Sum_{n,l >= 0} T(n,l)*x^n*y^l = sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))). (End)

Corrected by Petros Hadjicostas, May 23 2020

A098580 Expansion of (sqrt(1-8x)-4x)/sqrt(1-8*x).

Original entry on oeis.org

1, -4, -16, -96, -640, -4480, -32256, -236544, -1757184, -13178880, -99573760, -756760576, -5778898944, -44304891904, -340806860800, -2629081497600, -20331563581440, -157569617756160, -1223481737871360, -9515969072332800, -74124390668697600

Offset: 0

Paul Barry, Sep 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A059304.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((sqrt(1-8*x)-4*x)/sqrt(1-8*x))) // G. C. Greubel, Feb 03 2018

CoefficientList[Series[(Sqrt[1-8x]-4x)/Sqrt[1-8x],{x,0,20}],x] (* Harvey P. Dale, May 06 2017 *)

x='x+O('x^30); Vec((sqrt(1-8*x)-4*x)/sqrt(1-8*x)) \\ G. C. Greubel, Feb 03 2018

G.f.: 1-4*x/sqrt(1-8*x).
a(0)=1, a(n) = -2^(n+1)*(2*(n-1))!/(n-1)!^2.
D-finite with recurrence: (n-1)*a(n) +4*(3-2*n)*a(n-1)=0. - R. J. Mathar, Sep 26 2012

A103973 Expansion of (sqrt(1-8x^2)+8x^2+2x-1)/(2xsqrt(1-8x^2)).

Original entry on oeis.org

1, 2, 4, 4, 24, 16, 160, 80, 1120, 448, 8064, 2688, 59136, 16896, 439296, 109824, 3294720, 732160, 24893440, 4978688, 189190144, 34398208, 1444724736, 240787456, 11076222976, 1704034304, 85201715200, 12171673600, 657270374400

Offset: 0

Paul Barry, Feb 23 2005

Cf. A025225, A059304.

G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(2*x).
a(n) = sum{k=0..floor(n/2), 2^(n-k) * A000108(k) * C(k+1, n-k)}.
Conjecture D-finite with recurrence: 11*n*(n+1)*a(n)+4*n*(4*n+1)*a(n-1) +8*(27-11*n^2)*a(n-2) -32*(4*n+9)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 09 2012
a(n) ~ 2^((3*n + 1)/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^((3*n + 2)/2) / (sqrt(Pi)*n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021

A103978 Expansion of (sqrt(1-12x^2)+12x^2+2x-1)/(2xsqrt(1-12x^2)).

Original entry on oeis.org

1, 3, 6, 9, 54, 54, 540, 405, 5670, 3402, 61236, 30618, 673596, 288684, 7505784, 2814669, 84440070, 28146690, 956987460, 287096238, 10909657044, 2975361012, 124965162504, 31241290626, 1437099368796, 331638315876, 16581915793800

Offset: 0

Paul Barry, Feb 23 2005

Robert Israel, Table of n, a(n) for n = 0..1855

Cf. A025225, A059304, A103973.

rec:= -(n+1)*a(n)+2*(n-1)*a(n-1)+12*(2*n-3)*a(n-2)+24*(2-n)*a(n-3)+144*(4-n)*a(n-4):
f:= gfun:-rectoproc({rec=0,a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 9},a(n),remember):
map(f, [$0..30]); # Robert Israel, Sep 13 2020

CoefficientList[Series[(Sqrt[1-12x^2]+12x^2+2x-1)/(2x Sqrt[1-12x^2]),{x,0,30}],x] (* Harvey P. Dale, Aug 06 2022 *)

G.f.: 1/sqrt(1-12*x^2)+(1-sqrt(1-12*x^2))/(2*x).
a(n) = sum{k=0..floor(n/2), 3^(n-k) * A000108(k) * C(k+1, n-k)}.
D-finite with recurrence: -(n+1)*a(n)+2*(n-1)*a(n-1) +12*(2n-3)*a(n-2) +24(2-n)*a(n-3) + 144*(4-n)*a(n-4)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ 2^(n + 1/2) * 3^(n/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^(n + 1/2) * 3^((n+1)/2) / (sqrt(Pi) * n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021

A343842 Series expansion of 1/sqrt(8*x^2 + 1), even powers only.

Original entry on oeis.org

1, -4, 24, -160, 1120, -8064, 59136, -439296, 3294720, -24893440, 189190144, -1444724736, 11076222976, -85201715200, 657270374400, -5082890895360, 39392404439040, -305870434467840, 2378992268083200, -18531097667174400, 144542561803960320, -1128808577897594880

Offset: 0

Peter Luschny, May 04 2021

Essentially the inverse binomial convolution of the Delannoy numbers.

Signed version of A059304.

Cf. A008288, A006139.

gf := 1/sqrt(8*x^2 + 1): ser := series(gf, x, 32):
seq(coeff(ser, x, 2*n), n = 0..21);

Take[CoefficientList[Series[1/Sqrt[8*x^2 + 1], {x, 0, 42}], x], {1, -1, 2}] (* Amiram Eldar, May 05 2021 *)

my(x='x+O('x^25)); Vec(1/sqrt(8*x + 1)) \\ Michel Marcus, May 04 2021

a(n) = n! * [x^n] BesselJ(0, sqrt(8)*x).
D-finite with recurrence a(n) = 4*(1 - 2*n)*a(n - 1) / n for n >= 2.
a(n) = A(2*n) where A(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A008288(n, k).

cp's OEIS Frontend

A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

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A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

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A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.

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A386843 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k,n-k).

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A386918 a(n) = 2^n * binomial(4*n,n).

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A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2*a*x^2 + 1)^(n+1) from x = 0 to infinity.

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A098580 Expansion of (sqrt(1-8*x)-4*x)/sqrt(1-8*x).

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A261357 Pyramid of coefficients in expansion of (1 + 2x + 2y)^n.

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

A386843 a(n) = Sum_{k=0..n} binomial(2n+2,k) binomial(2*n-k,n-k).

A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.

A098580 Expansion of (sqrt(1-8x)-4x)/sqrt(1-8*x).

A103973 Expansion of (sqrt(1-8x^2)+8x^2+2x-1)/(2xsqrt(1-8x^2)).

A103978 Expansion of (sqrt(1-12x^2)+12x^2+2x-1)/(2xsqrt(1-12x^2)).