{"id":"n-70aa978122334d17","revision":1,"first_index":0,"fingerprint":"f6-10c18c8945c6e24a","name":"theta_plus_one_squared of Triangle of coefficients in expansion of (1+3*x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","12"],["1","24","81"],["1","36","243","432"],["1","48","486","1728","2025"],["1","60","810","4320","10125","8748"],["1","72","1215","8640","30375","52488","35721"],["1","84","1701","15120","70875","183708","250047","139968"],["1","96","2268","24192","141750","489888","1000188","1119744","531441"],["1","108","2916","36288","255150","1102248","3000564","5038848","4782969","1968300"],["1","120","3645","51840","425250","2204496","7501410","16796160","23914845","19683000","7144929"],["1","132","4455","71280","668250","4041576","16503102","46189440","87687765","108256500","78594219","25509168"],["1","144","5346","95040","1002375","6928416","33006204","110854656","263063295","433026000","471565314","306110016","89813529"],["1","156","6318","123552","1447875","11258676","61297236","240185088","683964567","1407334500","2043449694","1989715104","1167575877","312487308"],["1","168","7371","157248","2027025","17513496","107270163","480370176","1595917323","3940536600","7152073929","9285337152","8173031139","4374822312","1076168025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","106","712","4288","24064","128512","661504","3309568","16187392","77725696","367525888","1715470336","7918845952","36205232128"],"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"]],"recurrence":{"denominator":{"coeffs":[["1","-3","0"],["0","9","0"],["0","0","9"]]},"terms":[{"coeff":{"coeffs":[["1","9","27","27"],["0","9","45","54"],["0","0","9","27"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164754,"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-30efefd246e11889"]},{"id":164755,"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-30efefd246e11889"]},{"id":164756,"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-30efefd246e11889"]},{"id":164757,"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-30efefd246e11889"]}],"region_exact":{},"incoming_edges":[{"id":"e-30efefd246e11889","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-70aa978122334d17","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":"e0c21b68ca9785159e9cc58e35e85e849865f705d4cc2eda78b30c8e4fb4bfa1","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-6a5af5e1168bb679","role":"sequence","name":"Triangle of coefficients in expansion of (1+3*x)^n.","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;\n  1: 1,  12;\n  2: 1,  24,   81;\n  3: 1,  36,  243,   432;\n  4: 1,  48,  486,  1728,   2025;\n  5: 1,  60,  810,  4320,  10125,    8748;\n  6: 1,  72, 1215,  8640,  30375,   52488,   35721;\n  7: 1,  84, 1701, 15120,  70875,  183708,  250047,  139968;\n  8: 1,  96, 2268, 24192, 141750,  489888, 1000188, 1119744,  531441;\n  9: 1, 108, 2916, 36288, 255150, 1102248, 3000564, 5038848, 4782969, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = (1+(9*n+9)*x+(9*n^2+45*n+27)*x^2+(27*n^2+54*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*x + 9*n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(1 + (9*n - 3)*x + 9*n^2*x^2)); 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-6a5af5e1168bb679","steps":["(\u03b8+1)\u00b2"],"edges":["e-30efefd246e11889"],"text":"A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013610 \u2192 (\u03b8+1)\u00b2","chain":null}