cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A318397 Triangle read by rows: T(n,k) = binomial(n,k)^2 * binomial(2*(n-k), n-k).

Original entry on oeis.org

1, 2, 1, 6, 8, 1, 20, 54, 18, 1, 70, 320, 216, 32, 1, 252, 1750, 2000, 600, 50, 1, 924, 9072, 15750, 8000, 1350, 72, 1, 3432, 45276, 111132, 85750, 24500, 2646, 98, 1, 12870, 219648, 724416, 790272, 343000, 62720, 4704, 128, 1, 48620, 1042470, 4447872, 6519744, 4000752, 1111320, 141120, 7776, 162, 1
Offset: 0

Views

Author

Michael Somos, Aug 25 2018

Keywords

Comments

The square of the triangular matrix {binomial(n, k)^2}_{k=0..n} A008459 read by rows.

Examples

			Triangle begins:
   1
   2   1
   6   8   1
  20  54  18   1
  70 320 216  32   1
...

Crossrefs

Row sums give A002893.
Cf. A008459.

Programs

  • Mathematica
    T[ n_, k_] := Binomial[n, k]^2 Binomial[2 n - 2 k, n - k];
  • PARI
    {T(n, k) = binomial(n, k)^2 * binomial(2*(n-k), n-k)};

A338372 T(n, m) = Sum_{k=1..(m+3)/2} C(m-k+2, k-1)*C(n+1, k-1)*C(n-m+k-1, k-1)*C(2*n-2*k+4, 2*m-4*k+5)/(C(2*k-2, k-1)*C(2*m-2*k+4, 2*k-2))/2, triangle read by rows.

Original entry on oeis.org

1, 2, 2, 3, 10, 4, 4, 28, 32, 8, 5, 60, 136, 88, 16, 6, 110, 416, 504, 224, 32, 7, 182, 1036, 2024, 1616, 544, 64, 8, 280, 2240, 6448, 8064, 4736, 1280, 128, 9, 408, 4368, 17424, 31456, 28288, 13056, 2944, 256, 10, 570, 7872, 41616, 102592, 130880, 90880, 34432, 6656, 512, 11, 770, 13332, 90288, 291808, 501568, 487040, 273792, 87808, 14848, 1024
Offset: 0

Views

Author

Vladimir Kruchinin, Oct 23 2020

Keywords

Examples

			1,
2, 2,
3, 10, 4,
4, 28, 32, 8,
5, 60, 136, 88, 16

Crossrefs

Cf. A001263, A008459

Programs

  • Maple
    ogf := 1/(1 -2*x^2*y + x^2 - 2*x*y - 2*x): ser := series(ogf, x, 22):
xser := n -> expand(coeff(ser, x, n)):
seq(seq(coeff(xser(n), y, k), k=0..n), n=0..10); # Peter Luschny, Oct 23 2020
  • Maxima
    T(n,m):=sum((binomial(m-k+2,k-1)*binomial(n+1,k-1)*binomial(n-m+k-1,k-1)*binomial(2*n-2*k+4,2*m-4*k+5))/(binomial(2*k-2,k-1)*binomial(2*m-2*k+4,2*k-2)),k,1,(m+3)/2)/2;

Formula

G.f.: -1/(x^2*y^2 - (1 - x*(y + 2*A001263(x, y) + 1))^2) = 1/(1 - 2*x^2*y + x^2 - 2*x*y - 2*x).

A339000 Triangle read by rows: T(n, k) = C(n, k)*Sum_{j=0..n} C(n, k-j)*C(n+j, j)/C(2*j, j).

Original entry on oeis.org

1, 1, 2, 1, 7, 5, 1, 15, 32, 13, 1, 26, 111, 123, 34, 1, 40, 285, 603, 429, 89, 1, 57, 610, 2094, 2748, 1408, 233, 1, 77, 1155, 5845, 12170, 11196, 4437, 610, 1, 100, 2002, 14014, 42355, 60686, 42255, 13587, 1597, 1, 126, 3246, 30030, 124137, 254756, 271961, 150951, 40736, 4181
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 18 2020

Keywords

Examples

			Triangle begins as:
  1;
  1,  2;
  1,  7,   5;
  1, 15,  32,   13;
  1, 26, 111,  123,   34;
  1, 40, 285,  603,  429,   89;
  1, 57, 610, 2094, 2748, 1408, 233;

Links

Crossrefs

Cf. A000045 (Fibonacci), A001519, A008459, A046748 (row sums).

Programs

  • Magma
    b:=Binomial;
A339000:= func< n,k | b(n,k)*(&+[b(n,k-j)*b(n+j,j)/b(2*j,j): j in [0..n]]) >;
[A339000(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2024
  • Mathematica
    T[n_, k_]:= With[{B=Binomial}, B[n,k]*Sum[B[n,k-j]*B[n+j,j]/B[2*j,j], {j,0,n}]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)
  • Maxima
    T(n,m):=(binomial(n,m))*sum(((binomial(n,m-k))*(binomial(n+k,k)) )/(binomial(2*k,k)),k,0,n);
  • SageMath
    b=binomial
def A339000(n,k): return b(n,k)*sum(b(n,k-j)*b(n+j,j)//b(2*j,j) for j in range(n+1))
flatten([[A339000(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 31 2024

Formula

G.f.: A008459(x,y)/(1-x*y*A008459(x,y)^2).
T(n,n) = Fibonacci(2*n+1).

A385859 a(n) = Sum_{k=0..n} (C(n,k) mod 3)^2.

Original entry on oeis.org

1, 2, 6, 2, 4, 12, 6, 12, 21, 2, 4, 12, 4, 8, 24, 12, 24, 42, 6, 12, 21, 12, 24, 42, 21, 42, 66, 2, 4, 12, 4, 8, 24, 12, 24, 42, 4, 8, 24, 8, 16, 48, 24, 48, 84, 12, 24, 42, 24, 48, 84, 42, 84, 132, 6, 12, 21, 12, 24, 42, 21, 42, 66, 12, 24, 42, 24, 48, 84, 42
Offset: 0

Views

Author

Chai Wah Wu, Jul 10 2025

Keywords

Links

Crossrefs

Cf. A008459, A051638.

Programs

  • Mathematica
    a[n_]:=Sum[Mod[Binomial[n,k],3]^2,{k,0,n}]; Array[a,70,0] (* Stefano Spezia, Jul 10 2025 *)
  • Python
    from gmpy2 import digits
def A385859(n): return 5*3**(s:=digits(n,3)).count('2')-3<>1

Formula

If n has k '1' digits and m '2' digits in base 3, then a(n) = 2^(k-1)*(5*3^m - 3).

A039910 Lower triangular matrix T = Pascal lower triangular matrix divided on the left by its entry-square.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 12, -6, 1, 0, -132, 66, -12, 1, 0, 2280, -1140, 210, -20, 1, 0, -56760, 28380, -5240, 510, -30, 1, 0, 1923600, -961800, 177660, -17360, 1050, -42, 1, 0, -85149960, 42574980, -7864920, 769090, -46760, 1932, -56, 1
Offset: 0

Views

Author

Len Smiley

Keywords

Comments

Reference is a solution to a problem of I. Gessel.

Examples

			1;
0,1;
0,-2,1;
0,12,-6,1;
0,-132,66,-12,1;
etc.

References

  • "Inverse of a Combinatorial Matrix", Dave Callan, American Mathematical Monthly, Vol. 95 (1988), pp. 770-771.

Crossrefs

A007318, A008459.

Formula

T=Inverse matrix of {a_ij}:{binomial(i, j)*binomial(j, j-i)}=[ A008459 ]^(-1)*[ A007318 ]

A136100 Square each term in the sequence counting standard Young tableaux; cf. A117506.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 4, 9, 1, 1, 16, 25, 36, 25, 16, 1, 1, 25, 81, 25, 100, 256, 25, 100, 81, 25, 1, 1, 36, 196, 196, 225, 1225, 441, 441, 400, 1225, 196, 225, 196, 36, 1, 1, 49, 400, 784, 196, 441, 4096, 4900, 3136, 1764, 1225, 8100, 3136, 4900, 196
Offset: 0

Views

Author

Alford Arnold, Dec 27 2007

Keywords

Comments

When viewed as an irregular table there are A000041 entries per row which sum to 1, 1, 2, 6, 24, 120, 720 ...

Examples

			Row five of A117506 is 1 4 5 6 5 4 1 so row five of the present triangle is 1 16 25 36 25 16 1.

Crossrefs

Cf. A000041, A000085, A000142, A007318, A008459, A059797, A117506.

Formula

a(n) = A117506(n)*A117506(n).

Extensions

More terms from Alois P. Heinz, Nov 05 2015

A171605 Coefficients of Hankel moment polynomials for c=1/2:f(a,b) = Gamma[a + b]/Gamma[a] p(x,n)=Sum[Binomial(n, k)*(f(c, n)/(f(c, n - k)*f(c, k)))*x^k, {k, 0, n}].

Original entry on oeis.org

1, 1, -1, -1, 1, 1, 2, -17, 28, -17, 2, 1, 1, 9, -60, 116, -66, -66, 116, -60, 9, 1, 1, 20, -126, 196, 239, -1240, 1820, -1240, 239, 196, -126, 20, 1, 1, 35, -195, 15, 2205, -7001, 9785, -4845, -4845, 9785, -7001, 2205, 15, -195, 35, 1, 1, 54, -231, -880
Offset: 0

Views

Author

Roger L. Bagula, Dec 12 2009

Keywords

Comments

Row sums are zero except for n=0.
Other Hankel moments are:
c = 2 : A001263, Narayana
;c = 1 : A008459. binomial squared.

Examples

			{1},
{1, -1, -1, 1},
{1, 2, -17, 28, -17, 2, 1},
{1, 9, -60, 116, -66, -66, 116, -60, 9, 1},
{1, 20, -126, 196, 239, -1240, 1820, -1240, 239, 196, -126, 20, 1},
{1, 35, -195, 15, 2205, -7001, 9785, -4845, -4845, 9785, -7001, 2205, 15, -195, 35, 1},
{1, 54, -231, -880, 8052, -21912, 22276, 20976, -95634, 134596, -95634, 20976, 22276, -21912, 8052, -880, -231, 54, 1},
{1, 77, -182, -3094, 19929, -43043, -10920, 268568, -665406, 810810, -376740, -376740, 810810, -665406, 268568, -10920, -43043, 19929, -3094, -182, 77, 1},
{1, 104, 20, -7272, 37762, -42120, -270140, 1299080, -2608913, 2193808, 1776424, -7637904, 10518300, -7637904, 1776424, 2193808, -2608913, 1299080, -270140, -42120, 37762, -7272, 20, 104, 1},
{1, 135, 459, -13923, 55998, 59058, -1151070, 4053582, -6097509, -1814851, 27460881, -59839065, 67546644, -30260340, -30260340, 67546644, -59839065, 27460881, -1814851, -6097509, 4053582, -1151070, 59058, 55998, -13923, 459, 135, 1},
{1, 170, 1235, -23180, 59565, 411502, -3254225, 8979400, -4878915, -43714630, 159116983, -271019060, 204302345, 149970990, -623782445, 847660528, -623782445, 149970990, 204302345, -271019060, 159116983, -43714630, -4878915, 8979400, -3254225, 411502, 59565, -23180, 1235, 170, 1}

References

  • Philip Feinsilver and Rene Schott, Algebraic Structure and Operator Calculus; Volume I: Representations and Probability Theory,Kluwer,London,1993, ISBN 0-7923-2116-2,page 7

Crossrefs

A001263, A008459

Programs

  • Mathematica
    f[a_, b_] = Gamma[a + b]/Gamma[a]
c = 1/2;
p[x_, n_] = Sum[Binomial[n, k]*(f[c, n]/(f[ c, n - k]*f[c, k]))*x^k, {k, 0, n}]
Table[CoefficientList[p[x, n], x], {n, 0, 10}]
Flatten[%]

Formula

f(a,b) = Gamma[a + b]/Gamma[a]
p(x,n)=Sum[Binomial(n, k)*(f(c, n)/(f(c, n - k)*f(c, k)))*x^k, {k, 0, n}]

A178046 Triangle t(n, m) = 2*binomial(n,m)^2 -A008292(n+1,m+1)^2 read by rows.

Original entry on oeis.org

1, 1, 1, 1, -8, 1, 1, -103, -103, 1, 1, -644, -4284, -644, 1, 1, -3199, -91004, -91004, -3199, 1, 1, -14328, -1418031, -5836256, -1418031, -14328, 1, 1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1, 1, -251876
Offset: 0

Views

Author

Roger L. Bagula, May 18 2010

Keywords

Comments

Row sums are A028329(n) - A168562(n+1). - R. J. Mathar, Nov 05 2012

Examples

			1;
1, 1;
1, -8, 1;
1, -103, -103, 1;
1, -644, -4284, -644, 1;
1, -3199, -91004, -91004, -3199, 1;
1, -14328, -1418031, -5836256, -1418031, -14328, 1;
1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1;
1, -251876, -213392096, -7785232484, -24395306300, -7785232484, -213392096, -251876, 1;

Crossrefs

Cf. A177823, A008459, A141686, A008292

Programs

  • Mathematica
    << DiscreteMath`Combinatorica`
t[n_, m_] = 2*Binomial[n, m]^2 - Eulerian[n + 1, m]^2;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

A178343 Triangle T(n,m)= binomial(n, m)/Beta(m + 1, n - m + 1) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 12, 3, 4, 36, 36, 4, 5, 80, 180, 80, 5, 6, 150, 600, 600, 150, 6, 7, 252, 1575, 2800, 1575, 252, 7, 8, 392, 3528, 9800, 9800, 3528, 392, 8, 9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9, 10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10
Offset: 0

Views

Author

Roger L. Bagula, May 25 2010

Keywords

Comments

Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y) is the Beta-function.
Row sums are A037965(n+1). The second column is A011379.

Examples

			1;
2, 2;
3, 12, 3;
4, 36, 36, 4;
5, 80, 180, 80, 5;
6, 150, 600, 600, 150, 6;
7, 252, 1575, 2800, 1575, 252, 7;
8, 392, 3528, 9800, 9800, 3528, 392, 8;
9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9;
10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10;
11, 1100, 22275, 158400, 485100, 698544, 485100, 158400, 22275, 1100, 11;

Crossrefs

Cf. A037965

Programs

  • Mathematica
    Flatten[Table[Table[Binomial[n, m]/Beta[m + 1, n - m + 1], {m, 0, n}], {n, 0, 10}]]

Formula

T(n,m)=T(n,n-m) = (n+1)*( binomial(n,m))^2 = (n+1)*A008459(n).

Extensions

Edited by the Assoc. Eds. of the OEIS - Jun 27 2010

A338397 Triangle read by rows: T(n,m)= Sum_{k=0..m/2} C(n-k,m-2*k)*C(n-k,m-k)*C(n,k)/C(2*k,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 12, 4, 1, 16, 42, 34, 8, 1, 25, 110, 160, 90, 16, 1, 36, 240, 550, 540, 226, 32, 1, 49, 462, 1540, 2310, 1666, 546, 64, 1, 64, 812, 3724, 7910, 8596, 4802, 1282, 128, 1, 81, 1332, 8064, 23058, 34986, 29190, 13140, 2946, 256
Offset: 0

Views

Author

Vladimir Kruchinin, Oct 23 2020

Keywords

Examples

			1,
1, 1,
1, 4, 2,
1, 9, 12, 4,
1, 16, 42, 34, 8,
1, 25, 110, 160, 90, 16,
1, 36, 240, 550, 540, 226, 32

Crossrefs

Cf. A001263, A008459, A176280 (row sums), A338372.

Programs

  • Maxima
    T(n,m):=sum((binomial(n-k,m-2*k)*binomial(n-k,m-k)*binomial(n,k))/binomial(2*k,k) ,k,0,m/2);

Formula

G.f.: A008459(x,y)/(1-(x*y*A008459(x,y))^2).
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