A290595 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(16 - 24\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-352 - 3668\,x + 1680\,x^{2} + 1548\,n + 6024\,n x - 6264\,n x^{2} - 1683\,n^{2} - 5184\,n^{2} x + 7857\,n^{2} x^{2} + 648\,n^{3} + 2160\,n^{3} x - 4050\,n^{3} x^{2} - 81\,n^{4} - 324\,n^{4} x + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-303\,x - 937\,x^{2} + 1840\,x^{3} - 360\,n x - 2550\,n x^{2} - 6312\,n x^{3} + 621\,n^{2} x + 3780\,n^{2} x^{2} + 6939\,n^{2} x^{3} - 162\,n^{3} x - 1836\,n^{3} x^{2} - 2862\,n^{3} x^{3} + 324\,n^{4} x^{2} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(261\,x^{2} - 597\,x^{3} + 336\,x^{4} + 54\,n x^{2} + 1026\,n x^{3} - 1080\,n x^{4} - 81\,n^{2} x^{2} - 972\,n^{2} x^{3} + 1053\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-124 - 256\,x + 345\,n + 576\,n x - 261\,n^{2} - 432\,n^{2} x + 54\,n^{3} + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-39\,x + 64\,x^{2} - 90\,n x - 288\,n x^{2} + 54\,n^{2} x + 324\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | 8 | 9 | |||||||||||||
| 3 | 4 | 76 | 36 | 116 | ||||||||||||
| 4 | 28 | 888 | 1323 | 128 | 2367 | |||||||||||
| 5 | 280 | 12776 | 37152 | 14192 | 400 | 64800 | ||||||||||
| 6 | 3640 | 220096 | 1022094 | 840608 | 120875 | 1152 | 2208465 | |||||||||
| 7 | 58240 | 4429344 | 29419092 | 41002816 | 13892100 | 906012 | 3136 | 89710740 | ||||||||
| 8 | 1106560 | 102104768 | 904899708 | 1880287360 | 1135391000 | 190841400 | 6281947 | 8192 | 4220920935 | |||||||
| 9 | 24344320 | 2654422720 | 29931906384 | 85636988224 | 79734580000 | 25045725240 | 2329328680 | 41377216 | 20736 | 225398693520 | ||||||
| 10 | 608608000 | 76855645440 | 1065255473664 | 3975903159936 | 5224911959700 | 2609278142400 | 475151297250 | 26242459776 | 262856259 | 51200 | 13454469653625 | |||||
| 11 | 17041024000 | 2453123536640 | 40714741362816 | 190665529618560 | 333243452136000 | 240104804896656 | 71578204275900 | 8112544065024 | 279373430700 | 1625385500 | 123904 | 887170439855700 | ||||
| 12 | 528271744000 | 85587451317760 | 1666511741189952 | 9508547007734784 | 21181510912515000 | 20642232446762976 | 9063090411711168 | 1730710883854848 | 128307363093756 | 2853516119000 | 9841140419 | 294912 | 64009889847478575 | |||
| 13 | 17961239296000 | 3240693742917120 | 72822766142476800 | 494785459989499904 | 1360634224164915600 | 1713001031313882624 | 1033208140237528416 | 296426697957766656 | 38107860532740936 | 1916510154902600 | 28251869018736 | 58578611184 | 692224 | 5014189655924248800 | ||
| 14 | 664565853952000 | 132357858089676800 | 3386583576381225600 | 26902937411177897984 | 89091043139855078800 | 140018198981056336512 | 110452896665584777248 | 44182675410204269568 | 8702385165744770304 | 781311029492004800 | 27401625386008918 | 273076974762912 | 343793077147 | 1605632 | 423678728849595444225 | |
| 15 | 26582634158080000 | 5800376697904230400 | 167099441380203744000 | 1528964689011437705216 | 5977881113831358996800 | 11421442905478446598848 | 11361356492766855211392 | 6013238381486558241792 | 1675326621928460872896 | 234990490602706414400 | 15153522616615185572 | 378646806029082048 | 2590017648026068 | 1993719836668 | 3686400 | 38401661857252105912500 |
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Technical details
- Node id
n-3ca93ea710386bee· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-1c61ac6f25279709
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, 8; 3: 4, 76, 36; 4: 28, 888, 1323, 128; 5: 280, 12776, 37152, 14192, 400; 6: 3640, 220096, 1022094, 840608, 120875, 1152; 7: 58240, 4429344, 29419092, 41002816, 13892100, 906012, 3136; 8: 1106560, 102104768, 904899708, 1880287360, 1135391000, 190841400, 6281947, 8192; 9: 24344320, 2654422720, 29931906384, 85636988224, 79734580000, 25045725240, 2329328680, 41377216, 20736; 10: 608608000, 76855645440, 1065255473664, 3975903159936, 5224911959700, 2609278142400, 475151297250, 26242459776, 262856259, 51200;
%F FormulaThe row polynomials satisfy (9*n^2-24*n+16)*P(n,x) = -(81*n^4-648*n^3+1683*n^2-1548*n+352+(324*n^4-2160*n^3+5184*n^2-6024*n+3668)*x-(729*n^4-4050*n^3+7857*n^2-6264*n+1680)*x^2)*P(n-2,x) - ((162*n^3-621*n^2+360*n+303)*x-(324*n^4-1836*n^3+3780*n^2-2550*n-937)*x^2-(324*n^4-2862*n^3+6939*n^2-6312*n+1840)*x^3)*d/dx P(n-2,x) - ((81*n^2-54*n-261)*x^2-(324*n^3-972*n^2+1026*n-597)*x^3+(324*n^3-1053*n^2+1080*n-336)*x^4)*d^2/dx^2 P(n-2,x) + (54*n^3-261*n^2+345*n-124+(108*n^3-432*n^2+576*n-256)*x)*P(n-1,x) + ((54*n^2-90*n-39)*x-(108*n^3-324*n^2+288*n-64)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*P[n - 2, x] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*D[P[n - 2, x], {x, 2}] + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*P[n - 1, x] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 24*n + 16)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = (-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*v[i - 2] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*deriv(deriv(v[i - 2], x), x) + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*v[i - 1] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 24*n + 16)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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.