{"id":"n-8bb8b4f25546250f","revision":1,"first_index":0,"fingerprint":"f6-3f56d13b5db11e63","name":"theta_plus_one_squared of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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"],["4","4"],["13","32","9"],["40","168","108","16"],["121","736","783","256","25"],["364","2924","4464","2368","500","36"],["1093","10944","22086","16640","5625","864","49"],["3280","39376","99576","98720","47000","11448","1372","64"],["9841","137792","420642","521984","325375","110880","20923","2048","81"],["29524","472404","1693872","2539584","1976100","877716","230496","35328","2916","100"],["88573","1594336","6576471","11603712","10896375","6023808","2053884","436224","56133","4000","121"],["265720","5314424","24811524","50487600","55788000","37152864","15723120","4321536","767880","85000","5324","144"],["797161","17537568","91474137","211248640","269349750","210874752","107300004","36513792","8372565","1276000","123783","6912","169"],["2391484","57395644","330921504","856152448","1240111000","1119904056","669054624","273237504","77399388","15185500","2023120","174528","8788","196"],["7174453","186535808","1178403804","3379168768","5490132950","5632751808","3878632338","1857275904","630296073","152468800","26095586","3085056","239473","10976","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","54","332","1921","10656","57301","300836","1549566","7858040","39333637","194723136","954875233","4643959784","22422272022"],"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":[["0","0","0","0","0"],["128","32","32","0","0"],["0","0","0","0","0"],["0","0","-80","-40","0"],["0","0","0","0","0"],["0","0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["256","208","144","40","-8","0"],["256","320","128","-28","-44","0"],["0","112","-176","-208","-72","1"],["0","0","-160","-148","-36","3"],["0","0","0","-8","4","3"],["0","0","0","0","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["64","192","176","64","-24","-4","0"],["0","0","-112","-304","-188","-20","0"],["0","0","-112","-304","-184","-12","1"],["0","0","0","0","8","16","2"],["0","0","0","0","4","8","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165255,"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-ec090eec720f3720"]},{"id":165256,"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-ec090eec720f3720"]},{"id":165257,"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-ec090eec720f3720"]},{"id":165258,"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-ec090eec720f3720"]}],"region_exact":{},"incoming_edges":[{"id":"e-ec090eec720f3720","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-8bb8b4f25546250f","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":"1b9259fb22c2aeaf219009b75bb6c13c237eccad00f351a9d8f1847130a4c0c2","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-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:     4,      4;\n  2:    13,     32,       9;\n  3:    40,    168,     108,      16;\n  4:   121,    736,     783,     256,      25;\n  5:   364,   2924,    4464,    2368,     500,     36;\n  6:  1093,  10944,   22086,   16640,    5625,    864,     49;\n  7:  3280,  39376,   99576,   98720,   47000,  11448,   1372,    64;\n  8:  9841, 137792,  420642,  521984,  325375, 110880,  20923,  2048,   81;\n  9: 29524, 472404, 1693872, 2539584, 1976100, 877716, 230496, 35328, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (128*n+32*n*x-(80*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (256*n+256+(112*n^2+320*n+208)*x-(160*n^3+176*n^2-128*n-144)*x^2-(8*n^4+148*n^3+208*n^2+28*n-40)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (64+192*x-(112*n^2+112*n-176)*x^2-(304*n^2+304*n-64)*x^3+(4*n^4+8*n^3-184*n^2-188*n-24)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx 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[((256*n + 256 + (112*n^2 + 320*n + 208)*x - (160*n^3 + 176*n^2 - 128*n - 144)*x^2 - (8*n^4 + 148*n^3 + 208*n^2 + 28*n - 40)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (64 + 192*x - (112*n^2 + 112*n - 176)*x^2 - (304*n^2 + 304*n - 64)*x^3 + (4*n^4 + 8*n^3 - 184*n^2 - 188*n - 24)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(128*n + 32*n*x - (80*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)]];\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] = ((256*n + 256 + (112*n^2 + 320*n + 208)*x - (160*n^3 + 176*n^2 - 128*n - 144)*x^2 - (8*n^4 + 148*n^3 + 208*n^2 + 28*n - 40)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (64 + 192*x - (112*n^2 + 112*n - 176)*x^2 - (304*n^2 + 304*n - 64)*x^3 + (4*n^4 + 8*n^3 - 184*n^2 - 188*n - 24)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(128*n + 32*n*x - (80*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*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-6466a8550c08a52f","steps":["(\u03b8+1)\u00b2"],"edges":["e-ec090eec720f3720"],"text":"A116414 \u2192 (\u03b8+1)\u00b2"},"headline":"A116414 \u2192 (\u03b8+1)\u00b2","chain":null}