A105523 -id:A105523 - cp's OEIS frontend

A105522 Inverse of number triangle A105438.

1, -2, 1, 1, -2, 1, 2, 0, -2, 1, -2, 4, -1, -2, 1, -4, -1, 6, -2, -2, 1, 5, -10, 1, 8, -3, -2, 1, 10, 4, -18, 4, 10, -4, -2, 1, -14, 28, 0, -28, 8, 12, -5, -2, 1, -28, -14, 56, -8, -40, 13, 14, -6, -2, 1, 42, -84, -6, 96, -21, -54, 19, 16, -7, -2, 1, 84, 48, -180, 15, 150, -40, -70, 26, 18, -8, -2, 1, -132, 264, 33, -330, 55, 220, -66

Offset: 0

Paul Barry, Apr 11 2005

Row sums A105523 have g.f. 1-xc(-x^2) where c(x) is the g.f. of A000108. Diagonal sums are A105524.

			Rows begin {1}, {-2, 1}, {1, -2, 1}, {2, 0, -2, 1}, {-2, 4, -1, -2, 1}, ...

Riordan array ((1+2x+4x^2-(1+2x)sqrt(1+4x^2))/(2x^2), (sqrt(1+4x^2)-1)/(2x))

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

Original entry on oeis.org

1, -2, 1, 2, -3, 1, 0, 4, -4, 1, -2, -1, 7, -5, 1, 0, -4, -4, 11, -6, 1, 4, 2, -6, -10, 16, -7, 1, 0, 8, 8, -6, -20, 22, -8, 1, -10, -5, 11, 19, -1, -35, 29, -9, 1, 0, -20, -20, 7, 34, 13, -56, 37, -10, 1, 28, 14, -26, -46, -12, 49, 41, -84

Offset: 0

Paul Barry, Nov 01 2006

Inverse of triangle A106195.
Row sums are A105523 (expansion of 1-xc(-x^2) where c(x) is the g.f. of A000108).
Product of A007318 and A124448 is inverse of A053538.
A124448*A007318 = A106180, as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007
Triangle T(n,k), read by rows, given by (-2,1,-1,1,-1,1,-1,1,-1,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011

			Triangle begins
   1;
  -2,   1;
   2,  -3,   1;
   0,   4,  -4,   1;
  -2,  -1,   7,  -5,   1;
   0,  -4,  -4,  11,  -6,   1;
   4,   2,  -6, -10,  16,  -7,   1;
   0,   8,   8,  -6, -20,  22,  -8,   1;

Cf. A106195, A000045, A164948, A164942.

N=12;
T(n, k)=sum(i=0, n-k, binomial(k, i)*binomial(n-k, i)*2^(n-k-i));
M=matrix(N, N);
for(n=1, N, for(k=1, n, M[n, k]=T(n-1, k-1))); /* A106195 */
A=M^-1;  /* A124448 */
/* for (n=1, N, for(k=1, n, print1(M[n, k], ", "))); */ /* A106195 */
for (n=1, N, for(k=1, n, print1(A[n, k], ", "))); /* A124448 */
/* Joerg Arndt, May 14 2011 */

Edited by N. J. A. Sloane, Dec 29 2011

A125692 Riordan array (1-xc(-x^2),x(1-xc(-x^2))) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 1, 1, -3, 1, 0, 2, 3, -4, 1, -2, -2, 2, 6, -5, 1, 0, -4, -6, 0, 10, -6, 1, 5, 5, -3, -11, -5, 15, -7, 1, 0, 10, 15, 4, -15, -14, 21, -8, 1, -14, -14, 6, 26, 19, -15, -28, 28, -9, 1, 0, -28, -42, -16, 30, 42, -7, -48, 36, -10, 1

Offset: 0

Paul Barry, Nov 30 2006

Row sums are aerated signed Catalan numbers with g.f. c(-x^2). Inverse of A105306. First column is A105523.

			Triangle begins
1,
-1, 1,
0, -2, 1,
1, 1, -3, 1,
0, 2, 3, -4, 1,
-2, -2, 2, 6, -5, 1,
0, -4, -6, 0, 10, -6, 1
Contribution from _Paul Barry_, Apr 18 2010: (Start)
Production matrix begins
-1, 1,
-1, -1, 1,
-1, -1, -1, 1,
-1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, -1, -1, -1, -1, 1,
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1 (End)

A376134 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -1, -6, 17, 141, -660, -6688, 43837, 521755, -4412893, -60477282, 628119268, 9772644140, -120524236108, -2103803950976, 30068650440341, 582807287964375, -9477098158324107, -202143447363632090, 3686281848172281145, 85853256990102196221, -1735552985238117874788

Offset: 0

Ilya Gutkovskiy, Sep 11 2024

sign

Cf. A088716, A090192, A105523, A376135, A376137.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + x^2 A'[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A376135 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (2k+1) a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -2, -15, 86, 1030, -9844, -156219, 2098406, 41282298, -716119260, -16837011158, 358425572604, 9820300812556, -247923816153128, -7765514675946195, 226869417798485382, 8001626352728559218, -265582398152349968716, -10419379442081103988738

Offset: 0

Ilya Gutkovskiy, Sep 11 2024

sign

Cf. A000699, A090192, A105523, A376134, A376137.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + 2 x^2 A'[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A376137 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1)^2 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -3, -34, 495, 13631, -467404, -23984426, 1490938299, 123999435015, -12164649041259, -1497474725212924, 212746558833692052, 36393896155519042476, -7062273474686464802160, -1603475573855830444120802, 407344895625777134555939139, 118552169162473363108837155199, -38177398083353809033748641523305

Offset: 0

Ilya Gutkovskiy, Sep 11 2024

sign

Cf. A090192, A105523, A376095, A376134, A376135.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + 3 x^2 A'[-x] - x^3 A''[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-x) + 3 * x^2 * A'(-x) - x^3 * A''(-x)).

A291207 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + x/(1 - 2^kx/(1 + 3^kx/(1 - 4^kx/(1 + 5^kx/(1 - ...)))))).

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -3, 5, 0, 1, -1, -7, 27, 17, -2, 1, -1, -15, 167, 441, -121, 0, 1, -1, -31, 1071, 10673, -11529, -721, 5, 1, -1, -63, 6815, 262305, -1337713, -442827, 6845, 0, 1, -1, -127, 42687, 6525377, -161721441, -297209047, 23444883, 58337, -14

Offset: 0

Ilya Gutkovskiy, Aug 21 2017

			G.f. of column k: A_k(x) = 1 - x + (1 - 2^k)*x^2 + (2^(k + 1) - 4^k + 6^k - 1)*x^3 + ...
Square array begins:
   1,     1,       1,         1,           1,             1,  ...
  -1,    -1,      -1,        -1,          -1,            -1,  ...
   0,    -1,      -3,        -7,         -15,           -31,  ...
   1,     5,      27,       167,        1071,          6815,  ...
   0,    17,     441,     10673,      262305,       6525377,  ...
  -2,  -121,  -11529,  -1337713,  -161721441,  -19802585281,  ...

Columns k=0-2 give A105523, A202038, A193544.
Main diagonal gives A292920.
Cf. A290569.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-(-1)^i i^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 + x/(1 - 2^k*x/(1 + 3^k*x/(1 - 4^k*x/(1 + 5^k*x/(1 - ...)))))), a continued fraction.

A123254 Triangle T(n,k), 0<=k<=n, read by rows given by [ -1,1,-1,1,-1,1,-1,1,-1,1,...] DELTA [1,-1,1,-1,1,-1,1,-1,1,-1,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, 0, 0, 1, -3, 3, -1, 0, 0, 0, 0, 0, -2, 10, -20, 20, -10, 2, 0, 0, 0, 0, 0, 0, 0, 5, -35, 105, -175, 175, -105, 35, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 126, -504, 1176, -1764, 1764, -1176, 504, -126, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Oct 08 2006

			Triangle begins:
1;
-1, 1;
0, 0, 0;
1, -3, 3, -1;
0, 0, 0, 0, 0;
-2, 10, -20, 20, -10, 2;
0, 0, 0, 0, 0, 0, 0;
5, -35, 105, -175, 175, -106, 35, -5;
0, 0, 0, 0, 0, 0, 0, 0, 0;

Cf. A000108, A007318.

T(n,k)=(-1)^k*A105523(n)*binomial(n,k) . T(n,n)=A090192(n) . Sum_{k, 0<=k<=n}T(n,k)= 0^n= A000007(n).

cp's OEIS Frontend

A105522 Inverse of number triangle A105438.

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A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

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A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

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A125692 Riordan array (1-x*c(-x^2),x(1-x*c(-x^2))) where c(x) is the g.f. of A000108.

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A376134 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1) * a(k) * a(n-k-1).

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A376135 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (2*k+1) * a(k) * a(n-k-1).

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A376137 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1)^2 * a(k) * a(n-k-1).

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A291207 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + x/(1 - 2^k*x/(1 + 3^k*x/(1 - 4^k*x/(1 + 5^k*x/(1 - ...)))))).

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A123254 Triangle T(n,k), 0<=k<=n, read by rows given by [ -1,1,-1,1,-1,1,-1,1,-1,1,...] DELTA [1,-1,1,-1,1,-1,1,-1,1,-1,...] where DELTA is the operator defined in A084938.

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A125692 Riordan array (1-xc(-x^2),x(1-xc(-x^2))) where c(x) is the g.f. of A000108.

A376135 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (2k+1) a(k) * a(n-k-1).

A291207 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + x/(1 - 2^kx/(1 + 3^kx/(1 - 4^kx/(1 + 5^kx/(1 - ...)))))).