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.

Previous Showing 21-27 of 27 results.

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 06 2009

Keywords

Comments

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.
Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

Examples

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Crossrefs

Cf. A061554, A158793, A158815, A171224.

Formula

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.

A171486 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033321.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 21, 16, 9, 4, 1, 79, 58, 31, 14, 5, 1, 311, 224, 117, 52, 20, 6, 1, 1265, 900, 465, 205, 80, 27, 7, 1, 5275, 3720, 1910, 840, 330, 116, 35, 8, 1, 22431, 15713, 8034, 3532, 1396, 501, 161, 44, 9, 1, 96900, 67522, 34419, 15136, 6015, 2190
Offset: 0

Views

Author

Philippe Deléham, Dec 09 2009

Keywords

Comments

Equal to B*A065600 = A171224*B where B = A007318 ; equal to B*A039598*B^(-2).

Examples

			Triangle begins :
1
1, 1
2, 2, 1
6, 5, 3, 1
21, 16, 9, 4, 1
79, 58, 31, 14, 5, 1
311, 224, 117, 52, 20, 6, 1

Crossrefs

Cf. A171224, A104259, A091965

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n+1), A002212(n+1), A026378(n+1) for x = -1, 0, 1, 2, 3 respectively.
T(n,k) = T(n-1,k-1) + T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*2^i. - Philippe Deléham, Feb 23 2012

A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 6, 1, 5, 17, 35, 35, 10, 1, 6, 26, 75, 126, 96, 20, 1, 7, 37, 139, 339, 462, 267, 35, 1, 8, 50, 233, 758, 1558, 1716, 750, 70, 1, 9, 65, 363, 1491, 4194, 7247, 6435, 2123, 126, 1, 10, 82, 535, 2670, 9660, 23460, 34016, 24310, 6046, 252, 1, 11, 101, 755, 4451, 19846, 63195, 132339, 160795, 92378, 17303, 462
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A001405 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 2)*x^2/2! +  (k^3 + 3*k^2 + 6*k + 3)*x^3/3! + (k^4 + 4*k^3 + 12*k^2 + 12*k + 6)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   2,   5,   10,    17,    26,    37,  ...
   3,  13,   35,    75,   139,   233,  ...
   6,  35,  126,   339,   758,  1491,  ...
  10,  96,  462,  1558,  4194,  9660,  ...

Links

Crossrefs

Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.
Main diagonal gives A292631.

Programs

  • Maple
    [seq(seq((k)!*add((m-j)^(j-i)/floor(i/2)!/ceil(i/2)!/(j-i)!,i=0..j),j=0..m), m=0..20)]; # Robert Israel, Sep 20 2017
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x] (BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

A349540 E.g.f.: exp(x) * (BesselI(0,6*x) + BesselI(1,6*x)).

Original entry on oeis.org

1, 4, 25, 145, 931, 5866, 38359, 249880, 1655035, 10968724, 73320259, 491001721, 3304488565, 22283168350, 150744668065, 1021597533865, 6938921001235, 47202858834100, 321640950882475, 2194500145215595, 14992297096036345, 102535471011848230, 702004865920831525
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2021

Keywords

Crossrefs

Cf. A001405, A005773, A026378, A151318, A349541.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[x] (BesselI[0, 6 x] + BesselI[1, 6 x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] Binomial[k, Floor[k/2]] 3^k, {k, 0, n}], {n, 0, 22}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * binomial(k, k\2) * 3^k); \\ Michel Marcus, Nov 21 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(k,floor(k/2)) * 3^k.
a(n) ~ 7^(n + 1/2) / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 26 2021

A361964 Total number of peaks in 2-Fuss-skew paths of semilength n.

Original entry on oeis.org

2, 20, 226, 2696, 33138, 415164, 5270850, 67576208, 872918690, 11343392228, 148120453538, 1941910368280, 25545250484498, 337010368660876, 4457154741645954, 59076597464830240, 784518823873380930, 10435840680299248052, 139030100339736030306, 1854730153008453738408
Offset: 1

Views

Author

R. J. Mathar, Mar 31 2023

Keywords

Links

Crossrefs

Cf. A026378 (1-Fuss-skew), A361965 (3-Fuss-skew)

Programs

  • Maple
    FussSkewP := proc(l,n)
    local a,j,k ;
    a := 0 ;
    for j from 0 to n do
        a := a+sum( binomial(n,j) *binomial(j,k) *binomial(n*(l-1),n-2*j+k-1)
        * 2^(n*(l-2)+2*j-k+1)*3^(k-1)*(3*(n-j)+k),k=0..j) ;
    end do:
    a/n ;
end proc:
seq(FussSkewP(2,n),n=1..40) ;

Formula

D-finite with recurrence 2*n *(2*n-1) *(98653*n-203080) *a(n) +(-5301667*n^3 +13746049*n^2 -3506028*n -3685230) *a(n-1) +(-1931311*n^3 +43294062*n^2 -151212227*n +137614530) *a(n-2) +(n-3)*(8016735*n^2 -44290066*n +61812586) *a(n-3) +5*(n-3) *(n-4) *(129715*n-300617) *a(n-4)=0.

A361965 Total number of peaks in 3-Fuss-skew paths of semilength n.

Original entry on oeis.org

4, 96, 2672, 78848, 2400896, 74568704, 2347934464, 74675511296, 2393372833792, 77176031297536, 2500887165493248, 81372026697351168, 2656708513978580992, 86992366046604165120, 2855701159218522030080, 93950313500933860884480, 3096866628586659248603136
Offset: 1

Views

Author

R. J. Mathar, Mar 31 2023

Keywords

Links

Crossrefs

Cf. A026378 (1-Fuss-skew), A361964 (2-Fuss-skew)

Programs

  • Maple
    FussSkewP := proc(l,n)
    local a,j,k ;
    a := 0 ;
    for j from 0 to n do
        a := a+sum( binomial(n,j) *binomial(j,k) *binomial(n*(l-1),n-2*j+k-1)
        * 2^(n*(l-2)+2*j-k+1)*3^(k-1)*(3*(n-j)+k),k=0..j) ;
    end do:
    a/n ;
end proc:
seq(FussSkewP(3,n),n=1..40) ;

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 19 2009

Keywords

Examples

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Crossrefs

Cf. A107230, A171651
Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

Programs

  • Mathematica
    T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023
Previous Showing 21-27 of 27 results.

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