theta_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 2\,x + 4\,n x\right)\,P_{n}(x) = \left(-2\,x^{2} + 6\,x^{4} - 4\,x^{5} + 2\,n x^{2} - 8\,n x^{3} + 14\,n x^{4} + 4\,n x^{5} + 8\,n^{2} x^{3} - 20\,n^{2} x^{4} + 16\,n^{2} x^{5} - 16\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x^{2} - x^{3} + 3\,x^{4} + x^{5} - 2\,x^{6} - 6\,n x^{3} + 6\,n x^{5} - 8\,n^{2} x^{4} + 8\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(1 - 3\,x^{2} + 2\,x^{3} + 4\,n x + 12\,n x^{2} - 12\,n x^{3} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} - x^{3} + 2\,x^{4} + 4\,n x^{2} - 4\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 2\,x\)
\(P_{1}(x) = 1 + 6\,x + 9\,x^{2} + 4\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | x15 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | ||||||||||||||
| 1 | 1 | 6 | 9 | 4 | 20 | ||||||||||||
| 2 | 1 | 14 | 48 | 64 | 35 | 6 | 168 | ||||||||||
| 3 | 1 | 30 | 183 | 460 | 575 | 366 | 105 | 8 | 1728 | ||||||||
| 4 | 1 | 62 | 618 | 2504 | 5280 | 6336 | 4382 | 1648 | 279 | 10 | 21120 | ||||||
| 5 | 1 | 126 | 1977 | 11956 | 37770 | 70644 | 82418 | 60432 | 26901 | 6590 | 693 | 12 | 299520 | ||||
| 6 | 1 | 254 | 6156 | 53232 | 236745 | 630594 | 1082368 | 1236992 | 945891 | 473490 | 146388 | 24624 | 1651 | 14 | 4838400 | ||
| 7 | 1 | 510 | 18891 | 227740 | 1373385 | 4945986 | 11626699 | 18700056 | 21037563 | 16609570 | 9067641 | 3296124 | 740155 | 88158 | 3825 | 16 | 87736320 |
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Technical details
- Node id
n-dbfee4700b3112f1· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-d4a70096ed1391b3
Observed on the stored rows ?
- Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1, 2; 1: 1, 6, 9, 4; 2: 1, 14, 48, 64, 35, 6; 3: 1, 30, 183, 460, 575, 366, 105, 8; 4: 1, 62, 618, 2504, 5280, 6336, 4382, 1648, 279, 10; 5: 1, 126, 1977, 11956, 37770, 70644, 82418, 60432, 26901, 6590, 693, 12; 6: 1, 254, 6156, 53232, 236745, 630594, 1082368, 1236992, 945891, 473490, 146388, 24624, 1651, 14; 7: 1, 510, 18891, 227740, 1373385, 4945986, 11626699, 18700056, 21037563, 16609570, 9067641, 3296124, 740155, 88158, 3825, 16; 8: 1, 1022, 57486, 951448, 7592390, 35764908, 110230078, 235429232, 359861760, 399846400, 323715194, 188965848, 77490634, 21258692, 3567930, 306592, 8687, 18; 9: 1, 2046, 174045, 3917188, 40684220, 245107896, 962147116, 2629336160, 5204306358, 7635462340, 8399008574, 6939075144, 4272671260, 1924294232, 612769740, 130189504, 16648049, 1044270, 19437, 20;
%F FormulaThe row polynomials satisfy (1+(4*n-2)*x)*P(n,x) = ((2*n-2)*x^2+(8*n^2-8*n)*x^3-(20*n^2-14*n-6)*x^4-(16*n^3-16*n^2-4*n+4)*x^5)*P(n-2,x) - (x^2+(6*n+1)*x^3+(8*n^2-3)*x^4-(6*n+1)*x^5-(8*n^2-2)*x^6)*d/dx P(n-2,x) + (1+4*n*x+(12*n-3)*x^2+(16*n^2-12*n+2)*x^3)*P(n-1,x) + (x+(4*n-2)*x^2-x^3-(4*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = 1 + 6*x + 9*x^2 + 4*x^3.
%t MathematicaP[0, x_] := 1 + 2*x;
P[1, x_] := 1 + 6*x + 9*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*P[n - 2, x] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*D[P[n - 2, x], x] + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*P[n - 1, x] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = 1 + 6*x + 9*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*v[i - 2] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*deriv(v[i - 2], x) + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*v[i - 1] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
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Product of real-rooted factors (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
- ↑ Product of real-rooted factors (elementary) (above)
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.