A005260 -id:A005260 - cp's OEIS frontend

A202750 Triangle T(n,k) = binomial(n,k)^4 read by rows, 0<=k<=n.

1, 1, 1, 1, 16, 1, 1, 81, 81, 1, 1, 256, 1296, 256, 1, 1, 625, 10000, 10000, 625, 1, 1, 1296, 50625, 160000, 50625, 1296, 1, 1, 2401, 194481, 1500625, 1500625, 194481, 2401, 1, 1, 4096, 614656, 9834496, 24010000, 9834496, 614656, 4096, 1, 1, 6561, 1679616, 49787136, 252047376, 252047376, 49787136, 1679616, 6561, 1

Offset: 0

Charles R Greathouse IV, Dec 23 2011

Zhi-Wei Sun has conjectures related to the arithmetic mean of the polynomials formed from the rows of this sequence.

			Interpreted as polynomials:
1
x + 1
x^2 + 16*x + 1
x^3 + 81*x^2 + 81*x + 1
x^4 + 256*x^3 + 1296*x^2 + 256*x + 1
x^5 + 625*x^4 + 10000*x^3 + 10000*x^2 + 625*x + 1

Vincenzo Librandi, Rows n = 1..21, flattened
Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4p=x^2+dy^2 (2011).

Cf. A007318.

Row sums give A005260.

for(n=0,9,for(k=0,n,print1(binomial(n,k)^4", ")))

A216696 a(n) = Sum_{k=0..n} binomial(n,k)^4 * 2^k.

Original entry on oeis.org

1, 3, 37, 495, 7761, 131283, 2336629, 43174911, 819869185, 15906350403, 313905320037, 6281740700271, 127173173346129, 2599950664710675, 53601450936173877, 1113117091905581055, 23262762639358582785, 488890438209132473475, 10325711607889973605285, 219057502101780979753455

Offset: 0

Vaclav Kotesovec, Sep 15 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Cf. A005260.

Table[Sum[Binomial[n, k]^4*2^k, {k, 0, n}], {n, 0, 25}]

Recurrence: (864*n^8 + 14256*n^7 + 99843*n^6 + 386844*n^5 + 905129*n^4 + 1307419*n^3 + 1137462*n^2 + 545141*n + 110362)*a(n) - (673920*n^8 + 12130560*n^7 + 94006260*n^6 + 409620480*n^5 + 1097677875*n^4 + 1852470090*n^3 + 1922754750*n^2 + 1122222315*n + 281983230)*a(n+1) - (188352*n^8 + 3672864*n^7 + 30977310*n^6 + 147448176*n^5 + 432716089*n^4 + 800645440*n^3 + 910682766*n^2 + 581183533*n + 159056590)*a(n+2) - (10368*n^8 + 217728*n^7 + 1969236*n^6 + 10003440*n^5 + 31163253*n^4 + 60851106*n^3 + 72587550*n^2 + 48264909*n + 13672710)*a(n+3) + (864*n^8 + 19440*n^7 + 187539*n^6 + 1011492*n^5 + 3330104*n^4 + 6840009*n^3 + 8542572*n^2 + 5919152*n + 1739328)*a(n+4) = 0.
a(n) ~ (1+2^(1/4))^3/(4*2^(7/8)*Pi^(3/2)) * (1+2^(1/4))^(4*n)/n^(3/2). - Vaclav Kotesovec, Sep 19 2012
Generally, Sum_{k=0..n} binomial(n,k)^p*x^k is asymptotic a(n) ~ (1+x^(1/p))^(p*n+p-1)/sqrt((2*pi*n)^(p-1)*p*x^(1-1/p)). This is case p=4, x=2. - Vaclav Kotesovec, Sep 19 2012

Minor edits by Vaclav Kotesovec, Mar 31 2014

A096192 a(n) = Sum_{k=1..n} C(n,k)^4 where C(n,k) is binomial(n,k).

Original entry on oeis.org

1, 17, 163, 1809, 21251, 263843, 3395015, 44916497, 607041379, 8345319267, 116335834055, 1640651321763, 23365271704711, 335556407724359, 4854133484555663, 70666388112940817, 1034529673001901731

Offset: 1

Gerald McGarvey, Jul 25 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Equals A005260(n) - 1.

Table[Sum[Binomial[n, k-1]^4, {k, 0, n}], {n, 1, 25}] (* Vincenzo Librandi, May 03 2013 *)

a(n) ~ 2^(4*n + 1/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 27 2017

A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

Original entry on oeis.org

1, 3, 51, 1785, 67635, 2973753, 146591529, 7735733883, 430208938035, 24954576411225, 1496639801457801, 92241539987122683, 5816057121183700521, 373854785336483200155, 24431647104881328618315, 1619654401178752389082785

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 51*x^2/2!^4 + 1785*x^3/3!^4 + 67635*x^4/4!^4 +...
A(x)^(1/3) = 1 + x + x^2/2!^4 + x^3/3!^4 + x^4/4!^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002893, A005260, A141057, A180350, A336270.

{a(n)=if(n<0, 0, n!^4*polcoeff(sum(m=0, n, x^m/m!^4+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^4*sum(j=0, k, binomial(k, j)^4))}

a(n) = Sum_{k=0..n} C(n,k)^4 * Sum_{j=0..k} C(k,j)^4 = Sum_{k=0..n} C(n,k)^4 * A005260(k).

A178824 a(n) = Sum_{k=0..n} binomial(n,k)^4/(n+1).

Original entry on oeis.org

1, 1, 6, 41, 362, 3542, 37692, 424377, 4990722, 60704138, 758665388, 9694652838, 126203947828, 1668947978908, 22370427181624, 303383342784729, 4156846359584754, 57473870722327874, 801081711581734764, 11246487794657694810, 158920231643036635860, 2258896576436091238860

Offset: 0

Paul D. Hanna, Dec 27 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A000108, A005260, A131428.

List([0..20], n-> Sum([0..n], k-> Binomial(n,k)^4/(n+1) )); # G. C. Greubel, Jan 22 2019

[(&+[Binomial(n,k)^4/(n+1): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 22 2019

a:=n->add(binomial(n,k)^4/(n+1),k=0..n): seq(a(n),n=0..20); # Muniru A Asiru, Jan 22 2019

Table[Sum[Binomial[n,k]^4/(n+1), {k,0,n}], {n,0,20}] (* G. C. Greubel, Jan 22 2019 *)

{a(n)=sum(k=0, n, binomial(n, k)^4)/(n+1)}

[sum(binomial(n,k)^4/(n+1) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jan 22 2019

a(n) = A005260(n)/(n+1).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^2. - Seiichi Manyama, Mar 26 2025
a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 26 2025

A249332 a(n) = Sum_{k=0..2n} binomial(2n, k)^4.

Original entry on oeis.org

1, 18, 1810, 263844, 44916498, 8345319268, 1640651321764, 335556407724360, 70666388112940818, 15220552520052960516, 3337324864503769353060, 742446552655157791828680, 167167472732516775004539300, 38021985442071415426063237704, 8723111727436784830252513497928

Offset: 0

Michael Somos, Oct 25 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A001448, A005260, A006480, A050984.

[(&+[Binomial(2*n,k)^4: k in [0..2*n]]): n in [0..30]]; // G. C. Greubel, Aug 04 2018

Table[Sum[Binomial[2*n, k]^4, {k, 0, 2*n}], {n, 0, 30}] (* G. C. Greubel, Aug 04 2018 *)

{a(n) = sum(k=0, 2*n, binomial( 2*n, k)^4)};

a(n) = A005260(2*n).

A262705 Triangle: Newton expansion of C(n,m)^4, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 14, 1, 0, 36, 78, 1, 0, 24, 978, 252, 1, 0, 0, 4320, 8730, 620, 1, 0, 0, 8460, 103820, 46890, 1290, 1, 0, 0, 7560, 581700, 1159340, 185430, 2394, 1, 0, 0, 2520, 1767360, 13387570, 8314880, 595476, 4088, 1, 0, 0, 0, 3087000, 85806000, 170429490, 44341584, 1642788, 6552, 1

Offset: 0

Giuliano Cabrele, Sep 30 2015

Triangle here T_4(n,m) is such that C(n,m)^4 = Sum_{j=0..n} C(n,j)*T_4(j,m).
Equivalently, lower triangular matrix T_4 such that
|| C(n,m)^4 || = A202750 =  P * T_4 = A007318 * T_4.
T_4(n,m) = 0 for n < m and for 4*m < n.
Refer to comment to A262704.
Example:
C(x,2)^4 = x^4*(x-1)^4 /16 = 1*C(x,2) + 78*C(x,3) + 978*C(x,4) + 4320*C(x,5) + 8460*C(x,6) + 7560*C(x,7) + 2520*C(x,8);
C(5,2)^4 = C(5,3)^4 = 10000 = 1*C(5,2) + 78*C(5,3) + 978*C(5,4) + 4320*C(5,5) = 1*C(5,3) + 252*C(5,4) + 8730*C(5,5).

			Triangle starts:
[1];
[0,  1];
[0, 14,    1];
[0, 36,   78,      1];
[0, 24,  978,    252,     1];
[0,  0, 4320,   8730,   620,    1];
[0,  0, 8460, 103820, 46890, 1290, 1];

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

Row sums are, by definition, the inverse binomial transform of A005260.
Second diagonal (T_4(n+1,n)) is A058895(n+1).
Column T_4(n,2) is A122193(4,n).
Cf. A109983 (transpose of), A262704, A262706.
Cf. A078739, A078741.

[&+[(-1)^(n-j)*Binomial(n,j)*Binomial(j,m)^4: j in [0..n]]: m in [0..n], n in [0..10]]; // Bruno Berselli, Oct 01 2015

T4[n_, m_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[j, m]^4, {j, 0, n}]; Table[T4[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 01 2015 *)

// as a function
T_4:=(n,m)->_plus((-1)^(n-j)*binomial(n,j)*binomial(j,m)^4 $ j=0..n):
// as a matrix h x h
_P:=h->matrix([[binomial(n,m) $m=0..h]$n=0..h]):
_P_4:=h->matrix([[binomial(n,m)^4 $m=0..h]$n=0..h]):
_T_4:=h->_P(h)^-1*_P_4(h):

T_4(nmax) = {for(n=0, nmax, for(m=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n,j)*binomial(j,m)^4), ", ")); print())} \\ Colin Barker, Oct 01 2015

T_4(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^4.

Also, let S(r,s)(n,m) denote the Generalized Stirling2 numbers as defined in the link above, then T_4(n,m) = n! / (m!)^4 * S(m,m)(4,n).

A274786 Diagonal of the rational function 1/(1 - (wxz + wy + wz + xy + xz + y + z)).

Original entry on oeis.org

1, 6, 114, 2940, 87570, 2835756, 96982116, 3446781624, 126047377170, 4712189770860, 179275447715364, 6918537571788024, 270178056420497316, 10656693484898995800, 423937118582497715400, 16989669600664370275440, 685277433339552643145490, 27797911234749454227812460, 1133299570662800455270517700

Offset: 0

Gheorghe Coserea, Jul 14 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A000984, A005258, A262704, A268545 - A268555.

a[n_] := Sum[(-1)^j Binomial[2n, j] Binomial[j, n]^3, {j, n, 2n}];
(* or much faster *)
a[0] = 1; a[1] = 6; a[n_] := a[n] = (2*(2*n - 1)*(11*n^2 - 11*n + 3)*a[n - 1] + 4*(n - 1)*(2*n - 3)*(2*n - 1)*a[n - 2])/n^3;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *)

a(n) = sum(j=n, 2*n, (-1)^(j)*binomial(2*n, 2*n - j)*binomial(j, n)^3);

my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*z+w*y+w*z+x*y+x*z+y+z));
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(18, R, [x,y,z,w])

a(n) = Sum_{j=0..2*n} (-1)^j * binomial(2*n,j) * binomial(j,n)^3.
a(n) = T(2*n,n), where triangle T(n,k) is defined by A262704.
0 = (-x^2+44*x^3+16*x^4)*y''' + (-3*x+198*x^2+96*x^3)*y'' + (-1+144*x+108*x^2)*y' + (6+12*x)*y, where y is the g.f.
Recurrence: n^3*a(n) = 2*(2*n - 1)*(11*n^2 - 11*n + 3)*a(n-1) + 4*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Dec 01 2017
a(n) ~ 2^(2*n - 1) * phi^(5*n + 5/2) / (5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 01 2017
Conjecture: a(n) = [x^n] (1 + x)^(2*n) * P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. Cf. A005260(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x))^2, due to Carlitz. - Peter Bala, Sep 21 2021
a(n) = A000984(n) * A005258(n). - Peter Bala, Oct 12 2024

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

Column k=4 of A328748.
Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A126869 (m=2), A002898 (m=3), this sequence (m=4), A328751 (m=5).
Cf. A002899, A005260, A328725.

Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)

{a(n) = polcoef(polcoef(polcoef((-2+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = 2*(n-1)*n*(2*n - 1)*a(n-1) + 112*(n-1)^3*a(n-2) + 184*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 336*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n-4) * 7^(n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)

A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

Original entry on oeis.org

1, 1, 5, 37, 181, 1301, 9401, 65465, 498037, 3796021, 29221705, 230396585, 1828448425, 14651160265, 118544522045, 965075143037, 7907605360757, 65162569952245, 539515760866889, 4486877961224297, 37463151704756281, 313909383754331801, 2638892573249746445, 22249830926517611917

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).

Alois P. Heinz, Table of n, a(n) for n = 0..1053

Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.

Cf. A089627.

a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 07 2025

Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - w) - (x y)^2 z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 23}]

a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2025

a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - Alois P. Heinz, Apr 07 2025

cp's OEIS Frontend

A202750 Triangle T(n,k) = binomial(n,k)^4 read by rows, 0<=k<=n.

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A216696 a(n) = Sum_{k=0..n} binomial(n,k)^4 * 2^k.

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A096192 a(n) = Sum_{k=1..n} C(n,k)^4 where C(n,k) is binomial(n,k).

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A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

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A178824 a(n) = Sum_{k=0..n} binomial(n,k)^4/(n+1).

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A249332 a(n) = Sum_{k=0..2*n} binomial(2*n, k)^4.

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A262705 Triangle: Newton expansion of C(n,m)^4, read by rows.

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A249332 a(n) = Sum_{k=0..2n} binomial(2n, k)^4.

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.