Search: a158831 -id:a158831
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A158825
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Square array of coefficients in the successive iterations of x*C(x) = (1-sqrt(1-4*x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.
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+10
24
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1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 21, 14, 1, 5, 20, 54, 80, 42, 1, 6, 30, 110, 260, 322, 132, 1, 7, 42, 195, 640, 1310, 1348, 429, 1, 8, 56, 315, 1330, 3870, 6824, 5814, 1430, 1, 9, 72, 476, 2464, 9380, 24084, 36478, 25674, 4862, 1, 10, 90, 684, 4200, 19852, 67844, 153306, 199094, 115566, 16796
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OFFSET
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1,5
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LINKS
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FORMULA
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G.f. of column n = (g.f. of row n of A158830)/(1-x)^n.
Row k equals the first column of the k-th matrix power of Catalan triangle A033184; thus triangle A033184 transforms row n into row n+1 of this array (A158825). - Paul D. Hanna, Mar 30 2009
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EXAMPLE
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Square array of coefficients in iterations of x*C(x) begins:
1, 1, 2, 5, 14, 42, 132, 429, 1430, ... A000108;
1, 2, 6, 21, 80, 322, 1348, 5814, 25674, ... A121988;
1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, ... A158826;
1, 4, 20, 110, 640, 3870, 24084, 153306, 993978, ... A158827;
1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156, ... A158828;
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, ...;
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, ...;
1, 13, 182, 2639, 39130, 589680, 8999172, 138755799, 2157976392, ...;
1, 14, 210, 3255, 51520, 827960, 13464752, 221101608, 3660331064, ...;
1, 15, 240, 3960, 66640, 1137640, 19640032, 342179672, 6007747368, ...;
1, 16, 272, 4760, 84864, 1533672, 28012464, 516105720, 9578580504, ...;
ILLUSTRATE ITERATIONS.
Let G(x) = x*C(x), then the first few iterations of G(x) are:
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ...;
G(G(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + ...;
G(G(G(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 260*x^5 + ...;
G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 + ...;
...
RELATED TRIANGLES.
The g.f. of column n is (g.f. of row n of A158830)/(1-x)^n
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;
...
Triangle A158835 transforms one diagonal into the next:
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; ...
so that:
where the diagonals start:
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, ...].
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MATHEMATICA
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Clear[row]; nmax = 12;
row[n_]:= row[n]= CoefficientList[Nest[(1-Sqrt[1-4#])/2&, x, n] + O[x]^(nmax+1), x] //Rest;
T[n_, k_]:= row[n][[k]];
Table[T[n-k+1, k], {n, nmax}, {k, n}]//Flatten (* Jean-François Alcover, Jul 13 2018, updated Aug 09 2018 *)
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PROG
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(PARI) {T(n, k)= local(F=serreverse(x-x^2+O(x^(k+2))), G=x);
for(i=1, n, G=subst(F, x, G)); polcoeff(G, k)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A158835
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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).
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+10
12
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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
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OFFSET
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1,4
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LINKS
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EXAMPLE
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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:
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,...].
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.
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STATUS
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approved
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A158832
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Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
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+10
9
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1, 2, 12, 110, 1330, 19852, 351792, 7209036, 167607066, 4357308098, 125219900520, 3941126688798, 134808743674176, 4979127855477336, 197480359402576304, 8370550907396970684, 377599345119560766534, 18061714498169627460982
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]];
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A158833
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A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
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+10
6
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1, 3, 20, 195, 2464, 38052, 693048, 14528217, 344320262, 9100230282, 265305808404, 8456446272144, 292528760419440, 10913859037065560, 436812586581170976, 18668379209883807385, 848499254768957476312
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]];
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A158834
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A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
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+10
6
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1, 4, 30, 315, 4200, 67620, 1273668, 27454218, 666200106, 17968302638, 533188477536, 17261808531552, 605452449574320, 22870569475477112, 925663441858807096, 39964465820186753753, 1833332492818402014474
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]];
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A166905
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Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).
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+10
4
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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
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OFFSET
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0,4
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LINKS
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EXAMPLE
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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.
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PROG
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(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]}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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1, 4, 33, 380, 5510, 95732, 1933288, 44437080, 1144564278, 32638644236, 1020503373032, 34708182795156, 1275532011982176, 50365443858930384, 2126358227959866224, 95577781657788563192, 4556923094838105968302
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OFFSET
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0,2
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COMMENTS
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Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).
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LINKS
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PROG
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(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+4))), M, N, P, m=n); M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); N=matrix(m+2, m+2, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); P=matrix(m+2, m+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2, 2]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 9, 108, 1610, 28560, 586320, 13658904, 355787568, 10243342296, 322939137312, 11063339361360, 409194048521778, 16249995494795920, 689585033717023224, 31140529927119263136, 1490994828293677370148, 75444108490820383882392
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).
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LINKS
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PROG
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(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+5))), M, N, P, m=n); M=matrix(m+4, m+4, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); N=matrix(m+3, m+3, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); P=matrix(m+3, m+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 16, 270, 5148, 110929, 2677008, 71648322, 2107966432, 67649839664, 2352412120760, 88122951915388, 3538364803586104, 151611580761978784, 6905283671128114400, 333151832685664811338, 16973306740660778801468
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internal format)
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OFFSET
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0,2
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COMMENTS
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Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).
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LINKS
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PROG
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(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+6))), M, N, P, m=n); M=matrix(m+5, m+5, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+5)))); polcoeff(F, c)); N=matrix(m+4, m+4, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+5)))); polcoeff(F, c)); P=matrix(m+4, m+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4, 4]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 2, 11, 97, 1145, 16796, 293623, 5950792, 137075837, 3535416136, 100902444181, 3156570232069, 107392381479683, 3947409366073512, 155880018189733841, 6581149438442041483, 295807451972657856921, 14102499966460374953016
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).
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LINKS
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PROG
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(PARI) {a(n)=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+3)))); 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+3)))); polcoeff(F, c)); P=matrix(n+1, n+1, r, c, M[r+1, c]); M=(P~*N~^-1); sum(k=1, n+1, M[n+1, k])}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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