{"id":"n-426562d93f448042","revision":0,"first_index":0,"fingerprint":"f6-374be1e5539a9a21","name":"theta_plus_one_squared of shift_x_plus_one of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","8","9"],["2","20","36","16"],["4","48","117","96","25"],["8","112","342","400","200","36"],["16","256","936","1408","1025","360","49"],["32","576","2448","4480","4250","2196","588","64"],["64","1280","6192","13312","15500","10512","4165","896","81"],["128","2816","15264","37632","51800","43344","22638","7232","1296","100"],["256","6144","36864","102400","162400","161280","104272","44032","11745","1800","121"],["512","13312","87552","270336","484800","556416","428064","224256","79218","18100","2420","144"],["1024","28672","205056","696320","1392000","1810944","1613472","1007616","442260","134000","26741","3168","169"],["2048","61440","474624","1757184","3872000","5626368","5691840","4122624","2159784","814000","215622","38160","4056","196"],["4096","131072","1087488","4358144","10489600","16828416","19041792","15679488","9537264","4294400","1416184","332928","52897","5096","225"],["8192","278528","2469888","10649600","27788800","48761856","60988928","56229888","38918880","20363200","8028592","2351232","496522","71540","6300","256"],["16384","589824","5566464","25690112","72217600","137539584","188348160","192118784","149003712","88774400","40696656","14257152","3752476","718928","94725","7680","289"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["18","74","290","1098","4050","14634","52002","182250","631314","2165130","7361442","24839946","83259090","277412202","919392930"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","8","9"]],"recurrence":{"denominator":{"coeffs":[["4","30","48","24","4"],["0","6","20","18","4"],["0","0","1","2","1"]]},"terms":[{"coeff":{"coeffs":[["8","76","174","157","62","9"],["0","12","50","66","34","6"],["0","0","2","5","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174475,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-98d8f272b52b789c"]},{"id":174476,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-98d8f272b52b789c"]},{"id":174477,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-98d8f272b52b789c"]},{"id":174478,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-98d8f272b52b789c"]}],"region_exact":{},"incoming_edges":[{"id":"e-98d8f272b52b789c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-426562d93f448042","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"4cdea13a6f69190a2826ae4b1922e58ce1a4b40a73e455d59cf860ebbcc0479e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-21d2f90cb223c8f5","role":"sequence","name":"shift_x_plus_one of Riordan array (1-x,x/(1-x)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1,     8,     9;\n  1:   2,    20,    36,     16;\n  2:   4,    48,   117,     96,     25;\n  3:   8,   112,   342,    400,    200,     36;\n  4:  16,   256,   936,   1408,   1025,    360,     49;\n  5:  32,   576,  2448,   4480,   4250,   2196,    588,     64;\n  6:  64,  1280,  6192,  13312,  15500,  10512,   4165,    896,    81;\n  7: 128,  2816, 15264,  37632,  51800,  43344,  22638,   7232,  1296,   100;\n  8: 256,  6144, 36864, 102400, 162400, 161280, 104272,  44032, 11745,  1800,  121;\n  9: 512, 13312, 87552, 270336, 484800, 556416, 428064, 224256, 79218, 18100, 2420, 144;\n","recurrence_sentence":"The row polynomials satisfy (4+(6*n+30)*x+(n^2+20*n+48)*x^2+(2*n^2+18*n+24)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (8+(12*n+76)*x+(2*n^2+50*n+174)*x^2+(5*n^2+66*n+157)*x^3+(4*n^2+34*n+62)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 9*x^2.","mathematica":"P[0, x_] := 1 + 8*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + (12*n + 76)*x + (2*n^2 + 50*n + 174)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (4*n^2 + 34*n + 62)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(4 + (6*n + 30)*x + (n^2 + 20*n + 48)*x^2 + (2*n^2 + 18*n + 24)*x^3 + (n^2 + 4*n + 4)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (12*n + 76)*x + (2*n^2 + 50*n + 174)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (4*n^2 + 34*n + 62)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(4 + (6*n + 30)*x + (n^2 + 20*n + 48)*x^2 + (2*n^2 + 18*n + 24)*x^3 + (n^2 + 4*n + 4)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-34caf7309ed0b453","e-98d8f272b52b789c"],"text":"A159854 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A159854 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}