{"id":"n-57ebbee3d0c93bf9","revision":0,"first_index":0,"fingerprint":"f6-9974c1fe8b286c8c","name":"theta_plus_one_squared of Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","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","4"],["1","16","9"],["1","28","81","16"],["1","40","234","256","25"],["1","52","468","1120","625","36"],["1","64","783","3040","3875","1296","49"],["1","76","1179","6448","13825","10836","2401","64"],["1","88","1656","11776","36550","49392","26068","4096","81"],["1","100","2214","19456","80150","160056","148176","56064","6561","100"],["1","112","2853","29920","154750","417024","579474","389376","110565","10000","121"],["1","124","3573","43600","272500","934344","1775466","1812096","922185","203500","14641","144"],["1","136","4374","60928","447575","1874664","4583264","6465536","5043951","2008600","354046","20736","169"],["1","148","5256","82336","696175","3457980","10424162","19096064","20768319","12769900","4083508","587808","28561","196"],["1","160","6219","108256","1036525","5970384","21511931","48980096","69711759","60174400","29882281","7836192","938119","38416","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","126","556","2302","9108","34830","129708","472878","1694196","5982174","20863980","72000414","246194964"],"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"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["0","3","0","3","-12"],["0","2","15","152","88"],["1","-9","61","-114","0"],["0","9","-44","80","-128"],["0","0","9","-48","64"]]},"terms":[{"coeff":{"coeffs":[["0","1","4","36","32","-61","-12"],["0","2","9","30","-69","-12","40"],["0","1","17","-35","-145","162","0"],["0","0","9","-17","-104","240","-128"],["0","0","0","9","-57","112","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","44","210","676","256"],["0","0","54","272","292","48"],["1","2","17","68","-436","-384"],["0","9","-8","-68","-32","0"],["0","0","9","-30","-32","128"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173716,"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-0804d81549fd7d23"]},{"id":173717,"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-0804d81549fd7d23"]},{"id":173718,"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-0804d81549fd7d23"]},{"id":173719,"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-0804d81549fd7d23"]}],"region_exact":{},"incoming_edges":[{"id":"e-0804d81549fd7d23","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-57ebbee3d0c93bf9","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":"f100957c4b10f1b605cd228629894310a5f3afa0a37921df2c01e0f19d80fbb3","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-522def1e8ebf49cd","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","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,   4;\n  2: 1,  16,    9;\n  3: 1,  28,   81,    16;\n  4: 1,  40,  234,   256,    25;\n  5: 1,  52,  468,  1120,   625,     36;\n  6: 1,  64,  783,  3040,  3875,   1296,     49;\n  7: 1,  76, 1179,  6448, 13825,  10836,   2401,    64;\n  8: 1,  88, 1656, 11776, 36550,  49392,  26068,  4096,   81;\n  9: 1, 100, 2214, 19456, 80150, 160056, 148176, 56064, 6561, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^2+(9*n^3-9*n^2+2*n+3)*x+(9*n^4-44*n^3+61*n^2+15*n)*x^2-(48*n^4-80*n^3+114*n^2-152*n-3)*x^3+(64*n^4-128*n^3+88*n-12)*x^4)*P(n,x) = ((n^2+2*n+1)*x+(9*n^3+17*n^2+9*n+4)*x^2+(9*n^4-17*n^3-35*n^2+30*n+36)*x^3-(57*n^4+104*n^3+145*n^2+69*n-32)*x^4+(112*n^4+240*n^3+162*n^2-12*n-61)*x^5-(64*n^4+128*n^3-40*n+12)*x^6)*P(n-2,x) + (n^2+(9*n^3+2*n^2+2)*x+(9*n^4-8*n^3+17*n^2+54*n+44)*x^2-(30*n^4+68*n^3-68*n^2-272*n-210)*x^3-(32*n^4+32*n^3+436*n^2-292*n-676)*x^4+(128*n^4-384*n^2+48*n+256)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^2 + 2*n + 1)*x + (9*n^3 + 17*n^2 + 9*n + 4)*x^2 + (9*n^4 - 17*n^3 - 35*n^2 + 30*n + 36)*x^3 - (57*n^4 + 104*n^3 + 145*n^2 + 69*n - 32)*x^4 + (112*n^4 + 240*n^3 + 162*n^2 - 12*n - 61)*x^5 - (64*n^4 + 128*n^3 - 40*n + 12)*x^6)*P[n - 2, x] + (n^2 + (9*n^3 + 2*n^2 + 2)*x + (9*n^4 - 8*n^3 + 17*n^2 + 54*n + 44)*x^2 - (30*n^4 + 68*n^3 - 68*n^2 - 272*n - 210)*x^3 - (32*n^4 + 32*n^3 + 436*n^2 - 292*n - 676)*x^4 + (128*n^4 - 384*n^2 + 48*n + 256)*x^5)*P[n - 1, x])/(n^2 + (9*n^3 - 9*n^2 + 2*n + 3)*x + (9*n^4 - 44*n^3 + 61*n^2 + 15*n)*x^2 - (48*n^4 - 80*n^3 + 114*n^2 - 152*n - 3)*x^3 + (64*n^4 - 128*n^3 + 88*n - 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + 2*n + 1)*x + (9*n^3 + 17*n^2 + 9*n + 4)*x^2 + (9*n^4 - 17*n^3 - 35*n^2 + 30*n + 36)*x^3 - (57*n^4 + 104*n^3 + 145*n^2 + 69*n - 32)*x^4 + (112*n^4 + 240*n^3 + 162*n^2 - 12*n - 61)*x^5 - (64*n^4 + 128*n^3 - 40*n + 12)*x^6)*v[i - 2] + (n^2 + (9*n^3 + 2*n^2 + 2)*x + (9*n^4 - 8*n^3 + 17*n^2 + 54*n + 44)*x^2 - (30*n^4 + 68*n^3 - 68*n^2 - 272*n - 210)*x^3 - (32*n^4 + 32*n^3 + 436*n^2 - 292*n - 676)*x^4 + (128*n^4 - 384*n^2 + 48*n + 256)*x^5)*v[i - 1])/(n^2 + (9*n^3 - 9*n^2 + 2*n + 3)*x + (9*n^4 - 44*n^3 + 61*n^2 + 15*n)*x^2 - (48*n^4 - 80*n^3 + 114*n^2 - 152*n - 3)*x^3 + (64*n^4 - 128*n^3 + 88*n - 12)*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-522def1e8ebf49cd","steps":["(\u03b8+1)\u00b2"],"edges":["e-0804d81549fd7d23"],"text":"A209414 \u2192 (\u03b8+1)\u00b2"},"headline":"A209414 \u2192 (\u03b8+1)\u00b2","chain":null}