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.

Showing 1-10 of 15 results. Next

A085707 Triangular array A065547 unsigned and transposed.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 17, 17, 0, 1, 10, 55, 155, 155, 0, 1, 15, 135, 736, 2073, 2073, 0, 1, 21, 280, 2492, 13573, 38227, 38227, 0, 1, 28, 518, 6818, 60605, 330058, 929569, 929569, 0, 1, 36, 882, 16086, 211419, 1879038, 10233219, 28820619
Offset: 0

Views

Author

Philippe Deléham, Jul 19 2003

Keywords

Examples

			1;
1,  0;
1,  1,  0;
1,  3,  3,   0;
1,  6, 17,  17,   0;
1, 10, 55, 155, 155, 0;
...

References

  • Louis Comtet, Analyse Combinatoire, PUF, 1970, Tome 2, pp. 98-99.

Links

Crossrefs

Row sums Sum_{k>=0} T(n, k) = A006846(n), values of Hammersley's polynomial p_n(1).
Sum_{k>=0} 2^k*T(n, k) = A005647(n), Salie numbers.
Sum_{k>=0} 3^k*T(n, k) = A094408(n).
Sum_{k>=0} 4^k*T(n, k) = A000364(n), Euler numbers.
Cf. A006846, A005647, A000364, A065547, A000795.
Cf. A006846 A005647 A000364 A065547 A000795 A084938.

Programs

  • Mathematica
    h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First // Abs // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 02 2013 *)

Formula

Sum_{k >= 0} (-1/2)^k*T(n, k) = (1/2)^n.
Sum_{k >= 0} (-1/6)^k*T(n, k) = (4^(n+1)- 1)/3*(6^n).
Equals A000035 DELTA [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...], where DELTA is Deléham's operator defined in A084938.
T(n,n-1) = A110501(n), Genocchi numbers of first kind of even index. - Philippe Deléham, Feb 16 2007

A095652 Fourth column of Salié-triangle A065547.

Original entry on oeis.org

1, -6, 55, -736, 13573, -330058, 10233219, -393999940, 18443289777, -1031520165710, 67934648699151, -5203713576558056, 458704384454574365, -46104659405475675026, 5241286859201262537563, -669127269362962909934028, 95322254316544433430632009
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Crossrefs

Cf. A065547.

Programs

  • Mathematica
    col = 4; nmax = 16; h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x - 1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First; A065547 = Table[row[n], {n, 0, nmax + col}]; a[n_] := A065547[[n + col, col]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2013 *)

Formula

a(n) = A065547(n+3, 3), n>=0.

A095653 Fifth column of Salié-triangle A065547.

Original entry on oeis.org

1, -10, 135, -2492, 60605, -1879038, 72346915, -3386587600, 189409454649, -12474273613490, 955514574905471, -84228064915705380, 8465814535319723125, -962413843819623611302, 122866266360070054689435, -17503231485467828809214648, 2766579085410558809483880305
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Programs

  • Mathematica
    col = 5; nmax = 16; h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x - 1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First; A065547 = Table[row[n], {n, 0, nmax + col}]; a[n_] := A065547[[n + col, col]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2013 *)

Formula

a(n)= A065547(n+4, 4), n>=0.

A095654 Sixth column of Salié-triangle A065547.

Original entry on oeis.org

1, -15, 280, -6818, 211419, -8140201, 381046666, -21311675172, 1403560732197, -107511088907411, 9477041183699628, -952543229948336246, 108287370047760742687, -13824473678516613263949, 1969401123093628097438846, -311285602459098858965568152, 54311579841579438918099328713
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Formula

a(n)= A065547(n+5, 5), n>=0.

A095655 Seventh column of Salié-triangle A065547.

Original entry on oeis.org

1, -21, 518, -16086, 619455, -28997507, 1621813676, -106810679196, 8181571662285, -721200878840225, 72488343299017506, -8240646522324583186, 1052039595125648117163, -149871019205601062538351, 23688770132969478817542760, -4133100022172241089923256632
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Crossrefs

Cf. A065547.

Formula

a(n) = A065547(n+6, 6), n>=0.

A095656 Eighth column of Salié-triangle A065547.

Original entry on oeis.org

1, -28, 882, -34020, 1592811, -89086800, 5867161716, -449417759064, 39615920415765, -3981820505720948, 452662784400278870, -57789054676357516188, 8232489122499571570239, -1301235845855032664548680, 227033226003990207884798040, -43522196649760447692938214768
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Crossrefs

Cf. A065547.

Formula

a(n) = A065547(n+7, 7), n>=0.

A095657 Ninth column of Salié-triangle A065547.

Original entry on oeis.org

1, -36, 1410, -66132, 3699531, -243652552, 18663547812, -1645181597880, 165358218423381, -18798313917019868, 2399880945858805926, -341881241933228090220, 54038106872826560500575, -9428302924156701986153760, 1807402648330738178213456072
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Crossrefs

Cf. A065547.

Formula

a(n) = A065547(n+8, 8), n>=0.

A095658 Tenth column of Salié-triangle A065547.

Original entry on oeis.org

1, -45, 2145, -120219, 7919340, -606628022, 53474070910, -5374713460730, 611010175914787, -78004425857806195, 11112322069667692951, -1756425255264484313405, 306452433822808726723270, -58746833330940131294327300, 12321258284843928550078672460
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Crossrefs

Cf. A065547.

Formula

a(n) = A065547(n+9, 9), n>=0.

A095659 Eleventh column of Salié-triangle A065547.

Original entry on oeis.org

1, -55, 3135, -206921, 15853838, -1397545578, 140468598738, -15968808432126, 2038653078262911, -290421593787032345, 45904341056626435097, -8009163497468146614783, 1535354075146666869569332, -322017256862378971357996708, 73606894659465268081682059380
Offset: 0

Views

Author

Wolfdieter Lang, Jun 11 2004

Keywords

Formula

a(n) = A065547(n+10, 10), n>=0.

A098158 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 6, 1, 0, 0, 0, 5, 10, 1, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 0, 1, 66, 495, 924
Offset: 0

Views

Author

Paul Barry, Aug 29 2004

Keywords

Comments

Row sums are A011782. Inverse is A065547.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 29 2006
Sum of entries in column k is A001519(k+1) (the odd-indexed Fibonacci numbers). - Philippe Deléham, Dec 02 2008
Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k left-to-right minima. A left-to-right minimum in a permutation a(1)a(2)...a(n) is position i such that a(j) > a(i) for all j < i. - Tian Han, Nov 16 2023

Examples

			Rows begin
  1;
  0, 1;
  0, 1, 1;
  0, 0, 3, 1;
  0, 0, 1, 6, 1;

Links

Crossrefs

Cf. A098157, A034839.
Cf. A119900. - Philippe Deléham, Dec 02 2008

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 2*(n-k)) ))); # G. C. Greubel, Aug 01 2019
  • Magma
    [Binomial(n, 2*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019
  • Mathematica
    Table[Binomial[n, 2*(n-k)], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Oct 12 2016 *)
  • PARI
    {T(n,k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n), n, x) + y*O(y^k),k,y)} (Hanna)
  • PARI
    T(n,k) = binomial(n, 2*(n-k));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019
  • Sage
    [[binomial(n, 2*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

Formula

T(n,k) = binomial(n,2*(n-k)).
From Tom Copeland, Oct 10 2016: (Start)
E.g.f.: exp(t*x) * cosh(t*sqrt(x)).
O.g.f.: (1/2) * ( 1 / (1 - (1 + sqrt(1/x))*x*t) + 1 / (1 - (1 - sqrt(1/x))*x*t) ).
Row polynomial: x^n * ((1 + sqrt(1/x))^n + (1 - sqrt(1/x))^n) / 2. (End)
Column k is generated by the polynomial Sum_{j=0..floor(k/2)} C(k, 2j) * x^(k-j). - Paul Barry, Jan 22 2005
G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - Paul D. Hanna, Feb 25 2005
Sum_{k=0..n} x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 04 2006, Oct 15 2008, Oct 19 2008
T(n,k) = T(n-1,k-1) + Sum_{i=0..k-1} T(n-2-i,k-1-i); T(0,0)=1; T(n,k)=0 if n < 0 or k < 0 or n < k. E.g.: T(8,5) = T(7,4) + T(6,4) + T(5,3) + T(4,2) + T(3,1) + T(2,0) = 7+15+5+1+0+0 = 28. - Philippe Deléham, Dec 04 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively. - Philippe Deléham, Dec 24 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 14 2008
T(n,k) = A085478(k,n-k). - Philippe Deléham, Dec 02 2008
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 15 2012
Showing 1-10 of 15 results. Next

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