one_plus_derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(72 - 144\,n + 48\,n^{2}\right)\,P_{n}(x) = \left(-246 + 623\,x - 946\,x^{2} + 48\,x^{3} - 648\,x^{4} + 189\,x^{5} + 540\,n - 440\,n x + 1800\,n x^{2} - 910\,n x^{3} + 2604\,n x^{4} - 270\,n x^{5} - 192\,n^{2} + 96\,n^{2} x - 1000\,n^{2} x^{2} + 808\,n^{2} x^{3} - 2912\,n^{2} x^{4} + 144\,n^{2} x^{5} + 192\,n^{3} x^{2} - 192\,n^{3} x^{3} + 1248\,n^{3} x^{4} - 32\,n^{3} x^{5} - 192\,n^{4} x^{4}\right)\,P_{n-2}(x) + \left(-18 + 183\,x + 333\,x^{2} - 792\,x^{3} + 98\,x^{4} - 147\,x^{5} + 63\,x^{6} + 36\,n - 156\,n x - 460\,n x^{2} + 770\,n x^{3} - 164\,n x^{4} + 1074\,n x^{5} - 48\,n x^{6} + 48\,n^{2} x + 120\,n^{2} x^{2} - 224\,n^{2} x^{3} + 48\,n^{2} x^{4} - 784\,n^{2} x^{5} + 16\,n^{2} x^{6} + 160\,n^{3} x^{5}\right)\,P_{n-2}'(x) + \left(-6\,x + 103\,x^{2} - 30\,x^{3} - 94\,x^{4} + 36\,x^{5} - 9\,x^{6} + 12\,n x - 144\,n x^{2} - 12\,n x^{3} + 48\,n x^{4} + 96\,n x^{6} + 48\,n^{2} x^{2} - 16\,n^{2} x^{4} - 32\,n^{2} x^{6}\right)\,P_{n-2}''(x) + \left(258 - 37\,x - 354\,x^{2} + 63\,x^{3} - 540\,n - 236\,n x + 912\,n x^{2} - 48\,n x^{3} + 192\,n^{2} + 96\,n^{2} x - 784\,n^{2} x^{2} + 16\,n^{2} x^{3} + 192\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(6 - 31\,x + 36\,x^{2} - 81\,x^{3} - 12\,n + 240\,n x^{3} - 80\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 3 | 4 | 1 | 8 | ||||||||||||
| 2 | 7 | 26 | 28 | 10 | 1 | 72 | ||||||||||
| 3 | 15 | 108 | 251 | 256 | 117 | 20 | 1 | 768 | ||||||||
| 4 | 31 | 382 | 1526 | 2874 | 2856 | 1506 | 386 | 38 | 1 | 9600 | ||||||
| 5 | 63 | 1256 | 7773 | 23040 | 38222 | 37584 | 21906 | 7168 | 1155 | 72 | 1 | 138240 | ||||
| 6 | 127 | 3978 | 36072 | 155250 | 381627 | 582084 | 569416 | 356868 | 138405 | 30642 | 3312 | 138 | 1 | 2257920 | ||
| 7 | 255 | 12340 | 158719 | 946032 | 3226515 | 6963012 | 10000083 | 9760384 | 6465325 | 2838396 | 783597 | 123408 | 9345 | 268 | 1 | 41287680 |
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Technical details
- Node id
n-6dcdf88cf7bb3ab0· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-97627a1032bb0c6f
Observed on the stored rows ?
- Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
- 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; 1: 3, 4, 1; 2: 7, 26, 28, 10, 1; 3: 15, 108, 251, 256, 117, 20, 1; 4: 31, 382, 1526, 2874, 2856, 1506, 386, 38, 1; 5: 63, 1256, 7773, 23040, 38222, 37584, 21906, 7168, 1155, 72, 1; 6: 127, 3978, 36072, 155250, 381627, 582084, 569416, 356868, 138405, 30642, 3312, 138, 1; 7: 255, 12340, 158719, 946032, 3226515, 6963012, 10000083, 9760384, 6465325, 2838396, 783597, 123408, 9345, 268, 1; 8: 511, 37814, 676282, 5416282, 24607018, 71175174, 139486954, 191475770, 186729040, 129199090, 62362654, 20252766, 4148998, 480338, 26302, 526, 1; 9: 1023, 115008, 2823905, 29780464, 175398420, 656513648, 1677792268, 3052070640, 4044699674, 3944164720, 2826798342, 1470607952, 541090900, 134640336, 21049100, 1834192, 74367, 1040, 1;
%F FormulaThe row polynomials satisfy (48*n^2-144*n+72)*P(n,x) = -(192*n^2-540*n+246-(96*n^2-440*n+623)*x-(192*n^3-1000*n^2+1800*n-946)*x^2+(192*n^3-808*n^2+910*n-48)*x^3+(192*n^4-1248*n^3+2912*n^2-2604*n+648)*x^4+(32*n^3-144*n^2+270*n-189)*x^5)*P(n-2,x) + (36*n-18+(48*n^2-156*n+183)*x+(120*n^2-460*n+333)*x^2-(224*n^2-770*n+792)*x^3+(48*n^2-164*n+98)*x^4+(160*n^3-784*n^2+1074*n-147)*x^5+(16*n^2-48*n+63)*x^6)*d/dx P(n-2,x) + ((12*n-6)*x+(48*n^2-144*n+103)*x^2-(12*n+30)*x^3-(16*n^2-48*n+94)*x^4+36*x^5-(32*n^2-96*n+9)*x^6)*d^2/dx^2 P(n-2,x) + (192*n^2-540*n+258+(96*n^2-236*n-37)*x+(192*n^3-784*n^2+912*n-354)*x^2+(16*n^2-48*n+63)*x^3)*P(n-1,x) - (12*n-6+31*x-36*x^2+(80*n^2-240*n+81)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*P[n - 2, x] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*D[P[n - 2, x], x] + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*D[P[n - 2, x], {x, 2}] + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*P[n - 1, x] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*D[P[n - 1, x], x])/(48*n^2 - 144*n + 72)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*v[i - 2] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*deriv(v[i - 2], x) + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*deriv(deriv(v[i - 2], x), x) + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*v[i - 1] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*deriv(v[i - 1], x))/(48*n^2 - 144*n + 72)); 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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
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Product of real-rooted factors (elementary) polytool ? step ›
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
- ↑ Product of real-rooted factors (elementary) (above)
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from
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), …?
References
- [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.