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Search: a158825 -id:a158825
Displaying 1-10 of 23 results found. page 1 2 3
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A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108). +20
12
1, 1, 1, 4, 2, 1, 27, 11, 3, 1, 254, 94, 21, 4, 1, 3062, 1072, 217, 34, 5, 1, 45052, 15212, 2904, 412, 50, 6, 1, 783151, 257777, 47337, 6325, 695, 69, 7, 1, 15712342, 5074738, 906557, 116372, 12035, 1082, 91, 8, 1, 357459042, 113775490, 19910808, 2483706 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Paul D. Hanna, Table of n, a(n), n = 1..496 (rows 1..31).
EXAMPLE
Triangle T begins:
1;
1,1;
4,2,1;
27,11,3,1;
254,94,21,4,1;
3062,1072,217,34,5,1;
45052,15212,2904,412,50,6,1;
783151,257777,47337,6325,695,69,7,1;
15712342,5074738,906557,116372,12035,1082,91,8,1;
357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1;
9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1;
...
Array A158825 of coefficients in iterations of x*C(x) begins:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
1,9,90,945,10230,113190,1273668,14528217,167607066,...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,...;
...
This triangle transforms diagonals of A158825 into each other:
T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834;
where:
A158831 = [1,1,6,54,640,9380,163576,3305484,...];
A158832 = [1,2,12,110,1330,19852,351792,7209036,...];
A158833 = [1,3,20,195,2464,38052,693048,14528217,...];
A158834 = [1,4,30,315,4200,67620,1273668,27454218,...].
PROG
(PARI) {T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.
Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 28 2009, Mar 29 2009
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.
STATUS
approved
A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108). +20
10
1, 1, 6, 54, 640, 9380, 163576, 3305484, 75915708, 1952409954, 55573310936, 1734182983962, 58863621238500, 2159006675844616, 85088103159523296, 3585740237981536700, 160894462797493581048, 7658326127259130753070 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.
LINKS
Table of n, a(n) for n=1..18.
EXAMPLE
Table of coefficients in the i-th iteration of x*Catalan(x):
(1);
1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,(163576),1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,(3305484),31478628,303208212,...;
1,8,72,684,6720,67620,693048,7209036,(75915708),807845676,...;
1,9,90,945,10230,113190,1273668,14528217,167607066,(1952409954),...; ...
where terms in parenthesis form the initial terms of this sequence.
MATHEMATICA
nmax = 18;
g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];
T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];
Prepend[Diagonal[T, 1], 1] (* Jean-François Alcover, Jul 13 2018 *)
PROG
(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))), G=x); for(i=1, n-1, G=subst(F, x, G)); polcoeff(G, n)}
CROSSREFS
Cf. A158825, A158832, A158833, A158834.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108). +20
9
1, 2, 12, 110, 1330, 19852, 351792, 7209036, 167607066, 4357308098, 125219900520, 3941126688798, 134808743674176, 4979127855477336, 197480359402576304, 8370550907396970684, 377599345119560766534, 18061714498169627460982 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Triangle A158835 transforms A158831 into this sequence, where A158831 is the previous diagonal in A158825.
Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.
LINKS
Paul D. Hanna, Table of n, a(n), n = 1..50.
EXAMPLE
Array of coefficients in the i-th iteration of x*Catalan(x):
(1),1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,(2),6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,(12),54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,(110),640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,(1330),9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,(19852),163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,(351792),3305484,31478628,303208212,...;
1,8,72,684,6720,67620,693048,(7209036),75915708,807845676,...;
1,9,90,945,10230,113190,1273668,14528217,(167607066),...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...
where terms in parenthesis form the initial terms of this sequence.
MATHEMATICA
a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 18] (* Jean-François Alcover, Jul 13 2018, from PARI *)
PROG
(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))), G=x); for(i=1, n, G=subst(F, x, G)); polcoeff(G, n)}
CROSSREFS
Cf. A158825, A158831, A158833, A158834.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108). +20
6
1, 3, 20, 195, 2464, 38052, 693048, 14528217, 344320262, 9100230282, 265305808404, 8456446272144, 292528760419440, 10913859037065560, 436812586581170976, 18668379209883807385, 848499254768957476312 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Triangle A158835 transforms A158832 into this sequence, where A158832 is the previous diagonal in A158825.
Triangle A158835 transforms this sequence into A158834, the next diagonal in A158825.
LINKS
Table of n, a(n) for n=1..17.
EXAMPLE
Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
(1),2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,(3),12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,(20),110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,(195),1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,(2464),19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,(38052),351792,3305484,31478628,303208212,...;
1,8,72,684,6720,67620,(693048),7209036,75915708,807845676,...;
1,9,90,945,10230,113190,1273668,(14528217),167607066,...;
1,10,110,1265,14960,180510,2212188,27454218,(344320262),...;
1,11,132,1650,21164,276562,3666520,49181418,666200106,(9100230282),...; ...
where terms in parenthesis form the initial terms of this sequence.
MATHEMATICA
a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+1, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)
PROG
(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))), G=x); for(i=1, n+1, G=subst(F, x, G)); polcoeff(G, n)}
CROSSREFS
Cf. A158825, A158831, A158832, A158834.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
A158834 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108). +20
6
1, 4, 30, 315, 4200, 67620, 1273668, 27454218, 666200106, 17968302638, 533188477536, 17261808531552, 605452449574320, 22870569475477112, 925663441858807096, 39964465820186753753, 1833332492818402014474 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Triangle A158835 transforms A158833 into this sequence, where A158833 is the previous diagonal in A158825.
LINKS
Table of n, a(n) for n=1..17.
EXAMPLE
Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
(1),3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,(4),20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,(30),195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,(315),2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,(4200),38052,351792,3305484,31478628,303208212,...;
1,8,72,684,6720,(67620),693048,7209036,75915708,807845676,...;
1,9,90,945,10230,113190,(1273668),14528217,167607066,...;
1,10,110,1265,14960,180510,2212188,(27454218),344320262,...;
1,11,132,1650,21164,276562,3666520,49181418,(666200106),...;
1,12,156,2106,29120,409682,5841836,84218134,1225314662,(17968302638),...; ...
where terms in parenthesis form the initial terms of this sequence.
MATHEMATICA
a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+2, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)
PROG
(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))), G=x); for(i=1, n+2, G=subst(F, x, G)); polcoeff(G, n)}
CROSSREFS
Cf. A158825, A158831, A158832, A158833.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
A158829 Antidiagonal sums of square array A158825, in which row n lists the coefficients of the n-th iteration of x*C(x), where C(x) is the Catalan function (A000108). +20
4
1, 1, 2, 5, 15, 52, 202, 861, 3972, 19648, 103500, 577443, 3396804, 20988116, 135770140, 916936351, 6449233093, 47137434787, 357331341987, 2804582808108, 22754919576652, 190578011064394, 1645490708244886, 14629351150837605 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..24.
PROG
(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+1))), G=x, ADS=0); for(k=1, n, G=x; for(i=1, n-k, G=subst(F, x, G)); ADS=ADS+polcoeff(G, k)); ADS}
CROSSREFS
Cf. A158825, A158826, A158827, A158828.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
A158830 Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x). +20
4
1, 1, 0, 2, 0, 0, 5, 1, 0, 0, 14, 10, 0, 0, 0, 42, 70, 8, 0, 0, 0, 132, 424, 160, 4, 0, 0, 0, 429, 2382, 1978, 250, 1, 0, 0, 0, 1430, 12804, 19508, 6276, 302, 0, 0, 0, 0, 4862, 66946, 168608, 106492, 15674, 298, 0, 0, 0, 0, 16796, 343772, 1337684, 1445208, 451948 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Paul D. Hanna, Table of n, a(n), n = 1..1326 (rows 1..51).
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.
FORMULA
Row sums equal the factorial numbers.
G.f. of row n = (1-x)^n*[g.f. of column n of A158825] where the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108.
Row-reversal is triangle A122890 where g.f. of row n of A122890 = (1-x)^n*[g.f. of column n of A122888], and the g.f. of row n of array A122888 is the n-th iteration of x+x^2.
EXAMPLE
Triangle begins:
.1;
.1,0;
.2,0,0;
.5,1,0,0;
.14,10,0,0,0;
.42,70,8,0,0,0;
.132,424,160,4,0,0,0;
.429,2382,1978,250,1,0,0,0;
.1430,12804,19508,6276,302,0,0,0,0;
.4862,66946,168608,106492,15674,298,0,0,0,0;
.16796,343772,1337684,1445208,451948,33148,244,0,0,0,0;
.58786,1744314,10003422,16974314,9459090,1614906,61806,162,0,0,0,0;
.208012,8780912,71692452,180308420,161380816,51436848,5090124,103932,84,0,0,0,0;
....
where the g.f. of row n is (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of array A158825 is the n-th iteration of x*C(x):
.1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
.1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
.1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
.1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
.1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
.1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
.1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
.1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
....
ROW-REVERSAL yields triangle A122890:
.1;
.0,1;
.0,0,2;
.0,0,1,5;
.0,0,0,10,14;
.0,0,0,8,70,42;
.0,0,0,4,160,424,132;
.0,0,0,1,250,1978,2382,429;
.0,0,0,0,302,6276,19508,12804,1430; ...
where g.f. of row n = (1-x)^n*[g.f. of column n of A122888];
g.f. of row n of A122888 is the n-th iteration of x+x^2:
.1;
.1,1;
.1,2,2,1;
.1,3,6,9,10,8,4,1;
.1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1; ...
MATHEMATICA
nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand;
T = Table[SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x] // Reverse;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Oct 26 2018 *)
PROG
(PARI) {T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r, F=subst(F, x, CAT)); polcoeff(F, c)); Vec(truncate(Ser(vector(n+1, r, M[r, n+1])))*(1-x)^(n+1) +x*O(x^k))[k+1]}
CROSSREFS
Cf. A158825, A122890 (row-reversal), A122888, columns: A000108, A122892.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 28 2009
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.
STATUS
approved
A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108). +20
4
1, 1, 1, 6, 4, 1, 54, 33, 9, 1, 640, 380, 108, 16, 1, 9380, 5510, 1610, 270, 25, 1, 163576, 95732, 28560, 5148, 570, 36, 1, 3305484, 1933288, 586320, 110929, 13650, 1071, 49, 1, 75915708, 44437080, 13658904, 2677008, 353600, 31624, 1848, 64, 1, 1952409954 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..45.
EXAMPLE
Triangle begins:
1;
1,1;
6,4,1;
54,33,9,1;
640,380,108,16,1;
9380,5510,1610,270,25,1;
163576,95732,28560,5148,570,36,1;
3305484,1933288,586320,110929,13650,1071,49,1;
75915708,44437080,13658904,2677008,353600,31624,1848,64,1;
1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;
55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;
...
Coefficients in iterations of x*Catalan(x) form table A158825:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
...
This triangle T transforms rows into diagonals of A158825;
the initial diagonals begin:
A158831: [1,1,6,54,640,9380,163576,3305484,...];
A158832: [1,2,12,110,1330,19852,351792,7209036,...];
A158833: [1,3,20,195,2464,38052,693048,14528217,...];
A158834: [1,4,30,315,4200,67620,1273668,27454218,...].
For example:
T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;
T * [1,1,2,5,14,42,132,429,1430,...] = A158832;
T * [1,2,6,21,80,322,1348,5814, ...] = A158833;
T * [1,3,12,54,260, 1310, 6824, ...] = A158834.
PROG
(PARI) {T(n, k)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
CROSSREFS
Cf. A166906, A166907, A166908, A166909, variant: A166900.
Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 28 2009
STATUS
approved
A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1. +10
7
1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 10, 14, 0, 0, 0, 8, 70, 42, 0, 0, 0, 4, 160, 424, 132, 0, 0, 0, 1, 250, 1978, 2382, 429, 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430, 0, 0, 0, 0, 298, 15674, 106492, 168608, 66946, 4862, 0, 0, 0, 0, 244, 33148, 451948, 1445208, 1337684, 343772, 16796 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Main diagonal forms the Catalan numbers (A000108). Row sums gives the factorials. In table A122888, row n lists the coefficients of x^k, k = 1..2^n, in the n-th self-composition of (x + x^2) for n >= 0.
Parker gave the following combinatorial interpretation of the numbers: For n > 0, T(n, j) is the number of sequences c_1c_2...c_n of positive integers such that 1 <= c_i <= i for each i in {1, 2, .., n} with exactly j - 1 values of i such that c_i <= c_{i+1}. - Peter Luschny, May 05 2013
LINKS
Table of n, a(n) for n=0..65.
Toufik Mansour, Mark Shattuck, Statistics on bargraphs of inversion sequences of permutations, Discrete Math. Lett. (2020) Vol. 4, 42-49.
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.
Susan Field Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) (Section 2.3.4, p. 27,28.)
FORMULA
From Paul D. Hanna, Apr 11 2009: (Start)
G.f. of row n: (1-x)^n*[g.f. of column n of A122888] where the g.f. of row n of A122888 is the n-th iteration of x+x^2.
Row-reversal forms triangle A158830 where g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108. (End)
EXAMPLE
Triangle begins:
1;
0,1;
0,0,2;
0,0,1,5;
0,0,0,10,14;
0,0,0,8,70,42;
0,0,0,4,160,424,132;
0,0,0,1,250,1978,2382,429;
0,0,0,0,302,6276,19508,12804,1430;
0,0,0,0,298,15674,106492,168608,66946,4862;
0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;
0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;
0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,208012; ...
Table A122888 starts:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
where row n gives the g.f. of the n-th self-composition of (x+x^2).
From Paul D. Hanna, Apr 11 2009: (Start)
ROW-REVERSAL yields triangle A158830:
1;
1, 0;
2, 0, 0;
5, 1, 0, 0;
14, 10, 0, 0, 0;
42, 70, 8, 0, 0, 0;
132, 424, 160, 4, 0, 0, 0;
429, 2382, 1978, 250, 1, 0, 0, 0; ...
where
g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of A158825 = n-th iteration of x*Catalan(x).
RELATED ARRAY A158825 begins:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,...;
1,4,20,110,640,3870,24084,153306,993978,...;
1,5,30,195,1330,9380,67844,500619,3755156,...;
1,6,42,315,2464,19852,163576,1372196,11682348,...;
1,7,56,476,4200,38052,351792,3305484,31478628,...;
1,8,72,684,6720,67620,693048,7209036,75915708,...; ...
which consists of successive iterations of x*Catalan(x).
(End)
MATHEMATICA
nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand; T = Table[ SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x];
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)
CROSSREFS
Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.
Cf. related tables: A158830, A158825. [Paul D. Hanna, Apr 11 2009]
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Sep 18 2006
STATUS
approved
A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108). +10
6
1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, 1105478, 6227712, 35520498, 204773400, 1191572004, 6990859416, 41313818217, 245735825082, 1470125583756, 8840948601024, 53417237877396, 324123222435804, 1974317194619712 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Series reversion of x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8. - Benedict W. J. Irwin, Oct 19 2016
Column 1 of A106566^3 (see Barry, Section 3). - Peter Bala, Apr 11 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
FORMULA
a(n) = (1/n)*Sum_{k=1..n} [ binomial(2*k-2,k-1)*Sum_{i=k..n}( binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1) ) ]. - Vladimir Kruchinin, Jan 24 2013
G.f.: (1 - sqrt(-1 + 2*sqrt(-1 + 2*sqrt(1 - 4*x))))/2. - Benedict W. J. Irwin, Oct 19 2016
a(n) ~ 2^(8*n - 3) / (sqrt(5*Pi) * n^(3/2) * 39^(n - 1/2)). - Vaclav Kotesovec, Jul 20 2019
Conjecture D-finite with recurrence 1053*n*(n-1)*(n-2)*(n-3)*a(n) -36*(n-1)*(n-2)*(n-3)*(634*n-1367)*a(n-1) +24*(n-2)*(n-3)*(7966*n^2-43500*n+61181)*a(n-2) -8*(n-3)*(96128*n^3-957424*n^2+3221878*n-3665189)*a(n-3) +16*(91904*n^4-1446528*n^3+8575792*n^2-22703688*n+22652013)*a(n-4) -256*(8*n-35)*(8*n-41)*(8*n-39)*(8*n-37)*a(n-5)=0. - R. J. Mathar, Aug 30 2021
MATHEMATICA
max = 22; c[x_] := Sum[ CatalanNumber[n]*x^n, {n, 0, max}]; f[x_] := x*c[x]; CoefficientList[ Series[ f@f@f@x, {x, 0, max}], x] // Rest (* Jean-François Alcover, Jan 24 2013 *)
Rest@CoefficientList[InverseSeries[x-3x^2+6x^3-9x^4+10x^5-8x^6+4x^7-x^8+O[x]^30], x] (* Benedict W. J. Irwin, Oct 19 2016 *)
PROG
(PARI) a(n)=local(F=serreverse(x-x^2+O(x^(n+1))), G=x); for(i=1, 3, G=subst(F, x, G)); polcoeff(G, n)
(Maxima)
a(n):=sum(binomial(2*k-2, k-1)*sum(binomial(-k+2*i-1, i-1)*binomial(2*n-i-1, n-1), i, k, n), k, 1, n)/n; // Vladimir Kruchinin, Jan 24 2013
(Python)
from sympy import binomial as C
def a(n):
return sum(C(2*k - 2, k - 1) * sum(C(-k + 2*i - 1, i - 1) * C(2*n - i - 1, n - 1) for i in range(k, n + 1)) for k in range(1, n + 1)) / n
[a(n) for n in range(1, 51)] # Indranil Ghosh, Apr 12 2017
CROSSREFS
Cf. A121988 (2nd), A158825, A158827 (4th), A158828, A158829.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved
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Last modified August 30 04:38 EDT 2024. Contains 375526 sequences. (Running on oeis4.)