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 21-29 of 29 results.

A346078 G.f. A(x) satisfies: A(x) = 1 + x - x^2 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, -1, -2, -2, 1, 11, 33, 61, 22, -418, -2363, -8375, -19715, -6325, 263490, 1950298, 9423505, 33042827, 59212141, -283826231, -3970508822, -28167479326, -148668438363, -571280079455, -848399025239, 11052089847863, 148600718966518, 1198795581209734
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Crossrefs

Cf. A000587, A007476, A336970, A346079.

Programs

  • Mathematica
    nmax = 28; A[] = 0; Do[A[x] = 1 + x - x^2 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = -Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]

Formula

a(0) = a(1) = 1; a(n) = -Sum_{k=0..n-2} binomial(n-2,k) * a(k).

A171840 Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then the n-th row of the array = lim_{k->infinity}, k=1,2,3,...; (P(n))^k, deleting the first 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 4, 15, 1, 2, 4, 9, 52, 1, 2, 4, 8, 23, 203, 1, 2, 4, 8, 17, 65, 877, 1, 2, 4, 8, 16, 40, 199, 4140, 1, 2, 4, 8, 16, 33, 104, 654, 21147, 1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975, 1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570
Offset: 1

Views

Author

Gary W. Adamson, Dec 19 2009

Keywords

Comments

Row sums = A171841: (1, 3, 8, 22, 68, 241, 974, ...).
Right border = the Bell sequence A000110 starting (1, 2, 5, 15, 52, ...).
Row 2 of the array = A007476 starting (1, 1, 2, 4, 9, 23, 65, 199, ...).

Examples

			First few rows of the array:
  1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
  1, 1, 2,  4,  9,  23,  65,  199,   654, ...
  1, 1, 1,  2,  4,   8,  17,   40,   104, ...
  1, 1, 1,  1,  2,   4,   8,   16,    33, ...
  1, 1, 1,  1,  1,   2,   4,    8,    16, ...
  ...
Rightmost diagonal of 1's becomes leftmost column of the triangle:
  1;
  1, 2;
  1, 2, 5;
  1, 2, 4, 15;
  1, 2, 4,  9, 52;
  1, 2, 4,  8, 23, 203;
  1, 2, 4,  8, 17,  65, 877;
  1, 2, 4,  8, 16,  40, 199, 4140;
  1, 2, 4,  8, 16,  33, 104,  654, 21147;
  1, 2, 4,  8, 16,  32,  73,  291,  2296, 115975;
  1, 2, 4,  8, 16,  32,  65,  177,   857,   8569, 678570;
  ...
Example: n-th row corresponds to P(n) = Pascal's triangle with 1's column shifted up 1 row, so that P(1) =
  1;
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  ...
then take lim_{k->infinity} (P(1))^k, getting A000110: (1, 1, 2, 5, 15, 52, ...), then delete the first 1.

Crossrefs

Cf. A007318, A007476, A171841, A000110.

Programs

  • Sage
    # generates the diagonals of the triangle, starting with diag = 1 the Bell numbers.
def A171840_generator(len, diag) :
    A = [1]*diag
    for n in (0..len) :
        for k in range(n, 0, -1) :
            A[k - 1] += A[k]
        A.append(A[0])
        yield A[0]
for diag in (1..5) : print(list(A171840_generator(10, diag)))
# Peter Luschny, Feb 27 2012

Formula

Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then n-th row of the array = lim_{k->infinity} (P(n))^k, deleting the first 1.

A275872 A binomial convolution recurrence sequence.

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 18, 54, 173, 605, 2274, 9020, 37486, 163128, 743101, 3535765, 17518018, 90126158, 480514430, 2650912738, 15112253425, 88903779401, 539003066674, 3363608949132, 21581457167994, 142227480847092, 961868098767105, 6669657795455817, 47380035801732034, 344555811578909254, 2563218995058696890
Offset: 0

Views

Author

Olivier Gérard, Aug 11 2016

Keywords

Comments

Shifts 2 places left and decreases by one under a variant of binomial transform (see formula section).

Links

Crossrefs

Cf. A000994, A007476, A032346.

Programs

  • Maple
    A[0]:= 0:
A[1]:= 0:
for m from 2 to 50 do
  A[m]:= 1 + add(binomial(m-1,i+1)*A[i],i=0..m-2)
od:
seq(A[i],i=0..50); # Robert Israel, Aug 28 2016
  • Mathematica
    Clear[a]; a[0] = 0 ; a[1] = 0; a[n_] := a[n] = 1 + Sum[Binomial[n - 1, j+1]*a[j], {j, 0, n - 1}]; Table[a[n], {n, 0, 22}]
  • PARI
    first(n)=my(v=vector(n)); for(k=0,n-2, v[k+2]=sum(i=2,k, binomial(k+1,i+1)*v[i])+1); concat(0,v) \\ Charles R Greathouse IV, Aug 29 2016

Formula

Sum_{i=0..n} binomial(n+1,i+1)*a(i) = a(n+2) - 1.
G.f. g(x) satisfies g(x) = x^2/(1-x) + x^2*g(x/(1-x))/(1-x)^2. - Robert Israel, Aug 28 2016

A337186 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k).

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 36, 101, 308, 1013, 3562, 13300, 52482, 218045, 950614, 4335563, 20628882, 102153978, 525383324, 2801105889, 15455435864, 88117352141, 518391612686, 3142762585120, 19611454375090, 125829007917417, 829254498014570, 5608225148263459
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2021

Keywords

Crossrefs

Cf. A000994, A007476, A186021.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]

Formula

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (1 + x^2 * A(x/(1 - x))).

A351283 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^4.

Original entry on oeis.org

1, 1, 1, 5, 16, 46, 142, 496, 1888, 7538, 31291, 135739, 617461, 2939215, 14575027, 75014471, 399901294, 2205630124, 12572140372, 73961880118, 448447331338, 2798640572516, 17956583819425, 118336081817953, 800278211629795, 5549154792085813, 39420390891260821
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 12 2022

Keywords

Crossrefs

Cf. A007476, A045499, A351437, A351438.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - x)]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n + 1, k + 3] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n+1,k+3) * a(k).

A307875 G.f. A(x) satisfies: A(x) = 1 + x*(1 + x*A(x/(1 - x)^2)/(1 - x)^2).

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 78, 308, 1343, 6288, 31520, 169556, 976219, 5974381, 38597158, 262016556, 1864047379, 13870571346, 107732190252, 871392244426, 7323432971279, 63823847508765, 575833492837041, 5370836704825787, 51720220231890625, 513595474725399215, 5253091234710411001
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2019

Keywords

Crossrefs

Cf. A000110, A007476, A040027.

Programs

  • Mathematica
    terms = 26; A[] = 0; Do[A[x] = 1 + x (1 + x A[x/(1 - x)^2]/(1 - x)^2) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[Binomial[n + k - 1, 2 k + 1] a[k], {k, 0, n - 1}]; a[0] = 1; a[1] = 1; Table[a[n], {n, 0, 26}]

Formula

Recurrence: a(n+1) = Sum_{k=0..n} binomial(n+k,2*k+1)*a(k).

A331520 a(0) = a(1) = 1; a(n+2) = Sum_{k=0..n} (binomial(n,k) mod 2) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 3, 9, 7, 24, 8, 33, 17, 77, 27, 134, 66, 351, 67, 419, 135, 908, 204, 1469, 479, 3643, 553, 4572, 1182, 10227, 1889, 17125, 4641, 43640, 4642, 48283, 9285, 101211, 13929, 158786, 32504, 384441, 37153, 465259, 78957, 1020640, 125414, 1675453
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 19 2020

Keywords

Comments

Shifts 2 places left under the modulo 2 binomial transform.

Crossrefs

Cf. A007476, A010060, A047999, A166966.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = Sum[Mod[Binomial[n - 2, k], 2] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 47}]

Formula

a(n) = Sum_{k=0..n} (-1)^A010060(n-k) * (binomial(n, k) mod 2) * a(k+2).

A332885 a(0) = a(1) = 1; a(n) = a(n-2) + Sum_{k=0..n-2} binomial(n-2,k) * a(k).

Original entry on oeis.org

1, 1, 2, 3, 7, 16, 43, 123, 384, 1283, 4575, 17294, 69013, 289613, 1273934, 5856811, 28070535, 139936316, 724141487, 3882776711, 21536499372, 123388080843, 729195916303, 4439611287834, 27814781772073, 179132776279001, 1184720299683034, 8038979166269203
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2020

Keywords

Crossrefs

Cf. A000110, A007476, A040027.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = a[n - 2] + Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 (A[x] + (1/(1 - x)) A[x/(1 - x)]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x + x^2 * (A(x) + (1/(1 - x)) * A(x/(1 - x))).

A351648 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^5.

Original entry on oeis.org

1, 1, 1, 6, 22, 69, 224, 819, 3296, 13942, 60941, 276399, 1309207, 6479609, 33377271, 178186018, 983386188, 5604262733, 32955823822, 199771724691, 1246747659198, 8000380516898, 52728354046939, 356593588048023, 2472544614851517, 17563971319301049, 127727505109579581
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 16 2022

Keywords

Crossrefs

Cf. A007476, A045500, A351283, A351437, A351438.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - x)]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n + 2, k + 4] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n+2,k+4) * a(k).
Previous Showing 21-29 of 29 results.

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