Search: seq:1,1,1,1,2,1,1,3,3,2
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A340274
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Number of ways to write n as x + y + z with x, y, z positive integers such that 3*x^2*y^2 + 5*y^2*z^2 + 8*z^2*x^2 is a square.
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+30
5
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0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 4, 2, 5, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 3, 3, 4, 5, 2, 3, 5, 5, 4, 4, 2, 4, 4, 5, 3, 4, 3, 6, 3, 2, 5, 2, 2, 7, 7, 1, 3, 6, 4, 4, 3, 3, 6, 2, 5, 5, 3, 6, 5, 4, 6, 6, 6, 3, 6, 6, 4, 5, 6, 2, 6, 3, 5, 4, 5, 3, 5, 12, 4, 4, 5, 1, 6, 6, 7, 9, 3, 3, 6, 5, 6, 7, 4
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OFFSET
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1,9
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COMMENTS
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Conjecture 1: a(n) > 0 for all n > 2.
We have verified a(n) > 0 for all n = 3..10000. The conjecture holds if a(p) > 0 for every odd prime p. For any n > 0 we have a(3*n) > 0, since 3*n = n + n + n and 3 + 5 + 8 = 4^2.
It seems that a(n) = 1 only for n = 3..8, 10, 11, 19, 53, 89, 127, 178, 257, 461.
See also A343862 for similar conjectures.
Conjecture 1 holds for all n < 2^15. Note a(1823) = 1. - Martin Ehrenstein, May 03 2021
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LINKS
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EXAMPLE
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a(4) = 1 with 4 = 2 + 1 + 1 and 3*2^2*1^2 + 5*1^2*1^2 + 8*1^2*2^2 = 7^2.
a(19) = 1 with 19 = 9 + 9 + 1 and 3*9^2*9^2 + 5*9^2*1^2 + 8*1^2*9^2 = 144^2.
a(53) = 1 with 53 = 23 + 7 + 23 and 3*23^2*7^2 + 5*7^2*23^2 + 8*23^2*23^2 = 1564^2.
a(89) = 1 with 89 = 2 + 58 + 29 and 3*2^2*58^2 + 5*58^2*29^2 + 8*29^2*2^2 = 3770^2.
a(257) = 1 with 257 = 11 + 164 + 82 and 3*11^2*164^2 + 5*164^2*82^2 + 8*82^2*11^2 = 30340^2.
a(461) = 1 with 461 = 186 + 165 + 110 and 3*186^2*165^2 + 5*165^2*110^2 + 8*110^2*186^2 = 88440^2.
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MATHEMATICA
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SQ[n_]:=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[3x^2*y^2+(n-x-y)^2*(5*y^2+8*x^2)], r=r+1], {x, 1, n-2}, {y, 1, n-1-x}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
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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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A099509
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Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.
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+30
4
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1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 7, 1, 1, 5, 10, 16, 6, 3, 1, 6, 15, 30, 19, 16, 1, 1, 7, 21, 50, 45, 51, 10, 4, 1, 8, 28, 77, 90, 126, 45, 30, 1, 1, 9, 36, 112, 161, 266, 141, 126, 15, 5, 1, 10, 45, 156, 266, 504, 357, 393, 90, 50, 1, 1, 11, 55, 210, 414, 882, 784, 1016, 357
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OFFSET
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0,5
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COMMENTS
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Row sums form absolute values of A078039. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).
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LINKS
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FORMULA
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G.f.: (1-x+x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2+x^3*y^2+x^4*y^4).
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EXAMPLE
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Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,3,2],
[1,4,6,7,1],
[1,5,10,16,6,3],
[1,6,15,30,19,16,1],
[1,7,21,50,45,51,10,4],
[1,8,28,77,90,126,45,30,1],
[1,9,36,112,161,266,141,126,15,5],...
and can be derived from coefficients of (1+z+z^2)^n:
[1],
[1,1,1],
[1,2,3,2,1],
[1,3,6,7,6,3,1],
[1,4,10,16,19,16,10,4,1],
[1,5,15,30,45,51,45,30,15,5,1],...
by shifting each column k down by [k/2] rows.
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PROG
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(PARI) T(n, k)=if(n<k || k<0, 0, polcoeff((1+z+z^2+z*O(z^k))^(n-k\2), k, z))
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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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A368343
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Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).
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+30
4
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1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 7, 5, 2, 1, 5, 13, 16, 7, 2, 1, 6, 21, 41, 34, 9, 3, 1, 7, 31, 86, 125, 70, 12, 3, 1, 8, 43, 157, 346, 377, 143, 15, 3, 1, 9, 57, 260, 787, 1386, 1134, 289, 18, 4, 1, 10, 73, 401, 1562, 3937, 5547, 3405, 581, 22, 4
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OFFSET
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3,5
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LINKS
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FORMULA
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T(n,k) = T(n-3,k) + Sum_{j=0..n-3} k^j.
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k^2+k+1)) = Sum_{j=0..n} floor(k^j/(k^3-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - k*T(n-2,k) + T(n-3,k) - (k+1)*T(n-4,k) + k*T(n-5,k).
G.f. of column k: x^3/((1-x) * (1-k*x) * (1-x^3)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^3-1)) - floor((n+1)/3)) for k > 1.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 3, 7, 13, 21, 31, 43, ...
2, 5, 16, 41, 86, 157, 260, ...
2, 7, 34, 125, 346, 787, 1562, ...
2, 9, 70, 377, 1386, 3937, 9374, ...
3, 12, 143, 1134, 5547, 19688, 56247, ...
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PROG
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(PARI) T(n, k) = sum(j=0, n, k^(n-j)*(j\3));
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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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A247749
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Number T(n,k) of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y, consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3) for which the area below the path is k; triangle T(n,k), n>=0, read by rows.
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+30
3
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1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 2, 0, 1, 1, 4, 6, 6, 6, 3, 4, 2, 1, 1, 1, 5, 10, 13, 15, 12, 14, 15, 9, 12, 5, 5, 1, 1, 1, 6, 15, 24, 32, 33, 37, 46, 40, 43, 34, 28, 23, 16, 10, 5, 2, 1, 1, 7, 21, 40, 61, 75, 88, 114, 122, 134, 137, 118, 127, 101, 99, 69, 68, 41, 38, 19, 17, 5, 5, 0, 1
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OFFSET
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0,6
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LINKS
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FORMULA
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Sum_{k>=1} k * T(n,k) = A247748(n).
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EXAMPLE
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Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 1;
1, 3, 3, 2, 2, 0, 1;
1, 4, 6, 6, 6, 3, 4, 2, 1, 1;
1, 5, 10, 13, 15, 12, 14, 15, 9, 12, 5, 5, 1, 1;
1, 6, 15, 24, 32, 33, 37, 46, 40, 43, 34, 28, 23, 16, 10, 5, 2, 1;
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MAPLE
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b:= proc(x, y) option remember; `if`(y<0 or x<y, 0, `if`(x=0, 1,
expand(add(z^(y+j/2)*b(x-1, y+j), j=[-1, -3, 0, 1, 3]))))
end:
T:= n-> (p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
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MATHEMATICA
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b[x_, y_] := b[x, y] = If[y < 0 || x < y, 0, If[x == 0, 1,
Expand[Sum[z^(y+j/2)*b[x-1, y+j], {j, {-1, -3, 0, 1, 3}}]]]];
T[n_] := CoefficientList[b[n, 0], z];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A034929
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A triangle of Motzkin ballot numbers, read by rows.
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+30
1
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1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 4, 1, 5, 10, 13, 13, 9, 1, 6, 15, 24, 30, 30, 21, 1, 7, 21, 40, 59, 72, 72, 51, 1, 8, 28, 62, 105, 148, 178, 178, 127, 1, 9, 36, 91, 174, 276, 378, 450, 450, 323, 1, 10, 45, 128, 273, 480, 730, 980, 1158, 1158, 835, 1, 11, 55, 174, 410, 791
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OFFSET
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1,5
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COMMENTS
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REFERENCES
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M. Aigner, Motzkin numbers, Europ. J. Comb. 19 (1998), 663-675.
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LINKS
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FORMULA
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G.f.= 2(1+tz)/[1-2z+tz-2tz^2+sqrt(1-2tz-3t^2*z^2)].
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EXAMPLE
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Triangle begins:
[1],
[1, 1],
[1, 2, 1],
[1, 3, 3, 2],
[1, 4, 6, 6, 4],
[1, 5, 10, 13, 13, 9],
[1, 6, 15, 24, 30, 30, 21],
[1, 7, 21, 40, 59, 72, 72, 51]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 8, 4, 1, 5, 10, 20, 20, 9, 1, 6, 15, 40, 60, 54, 23, 1, 7, 21, 70, 140, 189, 161, 65, 1, 8, 28, 112, 280, 504, 644, 520, 199, 1, 9, 36, 168, 504, 1134, 1932, 2340, 1791, 654, 1, 10, 45, 240, 840, 2268, 4830, 7800, 8955, 6540, 2296
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OFFSET
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0,5
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COMMENTS
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Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199,...).
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LINKS
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FORMULA
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Triangle read by rows A007318 * (A007476 * 0^(n-k)) = binomial transform of an infinite lower triangular matrix with A007476 as the main diagonal: (1, 1, 1, 2, 4, 9, 23, 65, 199,...) and the rest zeros.
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EXAMPLE
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First few rows of the triangle =
1;
1, 1;
1, 2, 1;
1, 3, 3, 2;
1, 4, 6, 8, 4;
1, 5, 10, 20, 20, 9;
1, 6, 15, 40, 60, 54, 23;
1, 7, 21, 70, 140, 189, 161, 65;
1, 8, 28, 112, 280, 504, 644, 520, 199;
1, 9, 36, 168, 504, 1134, 1932, 2340, 1791, 654;
1, 10, 45, 240, 840, 2268, 4830, 7800, 8955, 6540, 2296;
...
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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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A144401
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Padovan ( A000931) version of A038137: expansion of polynomials as antidiagonal: p(x,n)=1/(1-x-x^3)^n.
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+30
0
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1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 3, 1, 5, 10, 13, 11, 4, 1, 6, 15, 24, 27, 18, 6, 1, 7, 21, 40, 55, 51, 30, 9, 1, 8, 28, 62, 100, 116, 94, 50, 13, 1, 9, 36, 91, 168, 231, 234, 171, 81, 19, 1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28, 1, 11, 55, 174, 402, 714, 987, 1065
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OFFSET
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1,5
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COMMENTS
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Row sums are: 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332 (cf. A008998).
These polynomials are sort of pseudo-combinations with the last element Padovan instead of one.
If you subtract the binomial triangle sequence you get:
{0},
{0, 0},
{0, 0, 0},
{0, 0, 0, 1},
{0, 0, 0, 2, 2},
{0, 0, 0, 3, 6, 3},
{0, 0, 0, 4, 12, 12, 5},
{0, 0, 0, 5, 20, 30, 23, 8},
{0, 0, 0, 6, 30, 60, 66, 42, 12}
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LINKS
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FORMULA
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p(x,n)=1/(1-x-x^3)^n; t(n,m)=anti_diagonal_expansion(p(x,n)).
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EXAMPLE
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{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 2},
{1, 4, 6, 6, 3},
{1, 5, 10, 13, 11, 4},
{1, 6, 15, 24, 27, 18, 6},
{1, 7, 21, 40, 55, 51, 30, 9},
{1, 8, 28, 62, 100, 116, 94, 50, 13},
{1, 9, 36, 91, 168, 231, 234, 171, 81, 19},
{1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28},
{1, 11, 55, 174, 402, 714, 987, 1065, 879, 527, 208, 41},
{1, 12, 66, 230, 585, 1152, 1792, 2220, 2175, 1640, 906, 330, 60},
{1, 13, 78, 297, 825, 1782, 3072, 4278, 4815, 4320, 3006, 1539, 520, 88},
{1, 14, 91, 376, 1133, 2662, 5028, 7752, 9807, 10122, 8391, 5424, 2586, 816, 129}
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MATHEMATICA
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Clear[f, b, a, g, h, n, t]; f[t_, n_] = 1/(1 - t - t^3)^n; a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b]
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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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A175424
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a(n) is the number of steps of iterations of {(((D_k^D_(k-1))^D_(k-2))^...)^D_1, where D_k is the k-th digit D of number r and k is the number of digits of number r in decimal expansion of r (A055642)} needed to reach a single-digit number starting at r = n, or a(n) = -1 if a single-digit number is never reached.
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+20
7
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, -1, -1, 3, 3, 2, 1, 1, 1, 4, 3, -1, -1, 3, 3, 3, 1, 1, 2, 2, 3, 3, 3, 2, 2, 2, 1, 1, 2, 3, 2, 4, 3, 2, 3, 2, 1, 1, 3, 3, 2, 3, 3, 3, 3, 2, 1, 1, 4, 3, 3, 3, 3, 3, 2, 3, 1, 1, 3, 2, 3, 2, 3, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 2, 2, 2
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OFFSET
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0,25
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COMMENTS
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Conjecture: max(a(n)) = 4.
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LINKS
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EXAMPLE
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For n = 33: a(33) = 4 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1}.
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PROG
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(PARI) f(n) = if (n, my(d=digits(n), r=d[#d]); if (!vecmin(d), return(0)); forstep (k=#d-1, 1, -1, r = r^d[k]; ); r); \\ A175420
findpos(n, list) = {forstep (k=#list, 1, -1, if (list[k] == n, return (k)); ); return (0); }
a(n) = {my(list = List(n), nb = 0); while (n >= 10, n = f(n); my(k=findpos(n, list)); nb++; if (k, if (k==#list-1, if (list[k]<10, return (nb), return(-1)), return(-1)); ); listput(list, n); ); return (nb); } \\ Michel Marcus, Jan 20 2022
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CROSSREFS
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KEYWORD
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sign,base
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 2, -1, 1, 3, -3, -2, 1, 4, -6, -8, 5, 1, 5, -10, -20, 25, 16, 1, 6, -15, -40, 75, 96, -61, 1, 7, -21, -70, 175, 336, -427, -272, 1, 8, -28, -112, 350, 896, -1708, -2176, 1385, 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936
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OFFSET
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0,5
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LINKS
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FORMULA
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|T(n, k)| = (-1)^(n - k) * n! * [x^(n - k)][y^n] (sec(y) + tan(y)) / exp(x*y).
T(n, k) = [x^(n - k)] -2^(k-(0^k))*(Euler(k, 0) + Euler(k, 1/2)) / (x-1)^(k + 1).
For a recursion see the Python program.
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EXAMPLE
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The triangle T(n, k) begins:
[0] 1;
[1] 1, 1;
[2] 1, 2, -1;
[3] 1, 3, -3, -2;
[4] 1, 4, -6, -8, 5;
[5] 1, 5, -10, -20, 25, 16;
[6] 1, 6, -15, -40, 75, 96, -61;
[7] 1, 7, -21, -70, 175, 336, -427, -272;
[8] 1, 8, -28, -112, 350, 896, -1708, -2176, 1385;
[9] 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936;
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MAPLE
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# Variant, computes abs(T(n, k)):
P := n -> n!*coeff(series((sec(y) + tan(y))/exp(x*y), y, 24), y, n):
seq(print(seq((-1)^(n - k)*coeff(P(n), x, n - k), k = 0..n)), n = 0..9);
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PROG
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(Python)
from functools import cache
@cache
def T(n: int, k: int) -> int:
if k == 0: return 1
if k == n:
p = k % 2
return p - sum(T(n, j) for j in range(p, n - 1, 2))
return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 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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