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 11-17 of 17 results.

A015121 Triangle of q-binomial coefficients for q=-9.

Original entry on oeis.org

1, 1, 1, 1, -8, 1, 1, 73, 73, 1, 1, -656, 5986, -656, 1, 1, 5905, 484210, 484210, 5905, 1, 1, -53144, 39226915, -352504880, 39226915, -53144, 1, 1, 478297, 3177326971, 257015284435, 257015284435, 3177326971, 478297, 1, 1, -4304672, 257363962948
Offset: 0

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Author

Olivier Gérard

Keywords

Comments

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former, or rows/columns of the latter, are: A000012 (k=0), A014991 (k=1), A015260 (k=2), A015277 (k=3), A015295 (k=4), A015315 (k=5), A015332 (k=6), A015349 (k=7), A015365 (k=8), A015381 (k=9), A015397 (k=10), A015414 (k=11), A015432 (k=12). - M. F. Hasler, Nov 05 2012

Links

Crossrefs

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Programs

  • Mathematica
    Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
  • PARI
    T015121(n, k, q=-9)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015123 Triangle of q-binomial coefficients for q=-10.

Original entry on oeis.org

1, 1, 1, 1, -9, 1, 1, 91, 91, 1, 1, -909, 9191, -909, 1, 1, 9091, 918191, 918191, 9091, 1, 1, -90909, 91828191, -917272809, 91828191, -90909, 1, 1, 909091, 9182728191, 917364637191, 917364637191, 9182728191, 909091, 1, 1, -9090909, 918273728191
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014992, A015261, A015278, A015298, A015316, A015333, A015350, A015367, A015382, A015398, A015417, A015433. - M. F. Hasler, Nov 04 & Nov 05 2012

Links

Crossrefs

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Programs

  • Mathematica
    Table[QBinomial[n, k, -10], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
  • PARI
    T015123(n, k, q=-10)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015124 Triangle of q-binomial coefficients for q=-11.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, 111, 111, 1, 1, -1220, 13542, -1220, 1, 1, 13421, 1637362, 1637362, 13421, 1, 1, -147630, 198134223, -2177691460, 198134223, -147630, 1, 1, 1623931, 23974093353, 2898705467483, 2898705467483, 23974093353, 1623931, 1, 1
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014993, A015262, A015279, A015300, A015317, A015334, A015353, A015368, A015383, A015499, A015418, A015434. - M. F. Hasler, Nov 04 & Nov 05 2012

Links

Crossrefs

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Programs

  • PARI
    T015124(n, k, q=-11)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015125 Triangle of q-binomial coefficients for q=-12.

Original entry on oeis.org

1, 1, 1, 1, -11, 1, 1, 133, 133, 1, 1, -1595, 19285, -1595, 1, 1, 19141, 2775445, 2775445, 19141, 1, 1, -229691, 399683221, -4793193515, 399683221, -229691, 1, 1, 2756293, 57554154133, 8283038077141, 8283038077141, 57554154133, 2756293, 1, 1
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former, or rows/columns of the latter, are, for k=0,...,12: A000012, A014994, A015264, A015281, A015302, A015319, A015336, A015354, A015369, A015384, A015401, A015421, A015436. - M. F. Hasler, Nov 04 2012

Links

Crossrefs

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Programs

  • PARI
    T015125(n, k, q=-12)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015132 Triangle of (Gaussian) q-binomial coefficients for q=-14.

Original entry on oeis.org

1, 1, 1, 1, -13, 1, 1, 183, 183, 1, 1, -2561, 36051, -2561, 1, 1, 35855, 7063435, 7063435, 35855, 1, 1, -501969, 1384469115, -19375002205, 1384469115, -501969, 1, 1, 7027567, 271355444571, 53166390519635, 53166390519635, 271355444571
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). - M. F. Hasler, Nov 04 2012

Links

Crossrefs

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015133 (q=-15). - M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Programs

  • PARI
    T015132(n, k, q=-14)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015196 Sum of Gaussian binomial coefficients for q=10.

Original entry on oeis.org

1, 2, 13, 224, 13435, 2266646, 1348019857, 2269339773068, 13484735901526279, 226960944509263279490, 13485189809930561625032701, 2269636415245291711513986785912, 1348523520252401463276762566348539123
Offset: 0

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

References

  • J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
  • M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Links

Crossrefs

Cf. A006116, A006117, A006118, A006119, A006120, A006121, A006122, A015195.
Row sums of triangle A022174.

Programs

  • Mathematica
    Total/@Table[QBinomial[n, m, 10], {n, 0, 20}, {m, 0, n}] (* Vincenzo Librandi, Nov 01 2012 *)
Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(10^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)

Formula

a(n) = 2*a(n-1)+(10^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013
a(n) ~ c * 10^(n^2/4), where c = EllipticTheta[3,0,1/10]/QPochhammer[1/10,1/10] = 1.348524024616... if n is even and c = EllipticTheta[2,0,1/10]/QPochhammer[1/10,1/10] = 1.2763120346269... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A275944 Gaussian binomial coefficient [n,3] for q = 10.

Original entry on oeis.org

1, 1111, 1122211, 1123333211, 1123445443211, 1123456666543211, 1123457788877543211, 1123457901110987543211, 1123457912334332087543211, 1123457913456666543087543211, 1123457913568899988653087543211, 1123457913580123333209753087543211, 1123457913581245667665420753087543211
Offset: 3

Views

Author

Ilya Gutkovskiy, Aug 13 2016

Keywords

Comments

More generally, the ordinary generation function for the Gaussian binomial coefficients [n,k]q is x^k/Product{m=0..k} (1 - q^m*x).
Convolution of A002275 and A147816 (considering offset: 0, 0, 1, 1100, 1110000, ...).
The first seven members are palindromes.

Links

Crossrefs

Cf. A002275, A022174, A109242, A147816.

Programs

  • Mathematica
    Table[((10^n - 100) (10^n - 10) (10^n - 1))/890109000, {n, 0, 15}]
Table[QBinomial[n, 3, 10], {n, 3, 15}]

Formula

O.g.f.: x^3/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
E.g.f.: (-1000 + 1110*exp(9*x) - 111*exp(99*x) + exp(999*x))*exp(x)/890109000.
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4).
a(n) = ((10^n - 100)*(10^n - 10)*(10^n - 1))/890109000.
a(n) = Product_{i=0..2} (1 - 10^(n-i))/(1 - 10^(i+1)).
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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