{"id":"n-e2ce22500c32dd0c","revision":1,"first_index":0,"fingerprint":"f6-b3fc9df967eda4ca","name":"theta_plus_one_squared of Pascal-(1,3,1) array.","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","20","9"],["1","36","81","16"],["1","52","297","208","25"],["1","68","657","1168","425","36"],["1","84","1161","3920","3225","756","49"],["1","100","1809","9488","14825","7236","1225","64"],["1","116","2601","18896","48025","42516","14161","1856","81"],["1","132","3537","33168","122025","175716","101577","25152","2673","100"],["1","148","4617","53328","262425","558900","514353","213312","41553","3700","121"],["1","164","5841","80400","501225","1463940","1992585","1283136","407025","64900","4961","144"],["1","180","7209","115408","876825","3319380","6267345","5901120","2840913","721300","96921","6480","169"],["1","196","8721","159376","1434025","6747300","16763145","21894720","15181425","5736100","1205281","139536","8281","196"],["1","212","10377","213328","2224025","12600180","39501105","68347200","65297745","35000500","10764281","1919952","194857","10388","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","134","583","2355","9196","34748","128253","464081","1652458","5804322","20153251","69278303","236084376"],"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"],["-1","11","5","49","0"],["-1","2","40","94","-7"],["0","-12","4","76","-4"],["0","0","-16","16","0"]]},"terms":[{"coeff":{"coeffs":[["0","28","58","378","190","-14"],["-1","7","231","499","374","-22"],["-1","-30","-80","-42","289","-8"],["0","-12","-156","-356","76","0"],["0","0","-16","-112","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","21","146","658","-51","-7"],["0","1","37","422","1398","-55","-11"],["0","0","12","240","1016","16","-4"],["0","0","0","16","224","16","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163866,"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-80bca760e085f131"]},{"id":163867,"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-80bca760e085f131"]},{"id":163868,"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-80bca760e085f131"]},{"id":163869,"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-80bca760e085f131"]}],"region_exact":{},"incoming_edges":[{"id":"e-80bca760e085f131","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-e2ce22500c32dd0c","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":"b29b92189b62a24170adcbcdb26af2311ffe5e838ec7d73c484cb3b8f7e96123","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-d8f9ad9ce0e039d0","role":"sequence","name":"Pascal-(1,3,1) array.","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,  20,    9;\n  3: 1,  36,   81,    16;\n  4: 1,  52,  297,   208,     25;\n  5: 1,  68,  657,  1168,    425,     36;\n  6: 1,  84, 1161,  3920,   3225,    756,     49;\n  7: 1, 100, 1809,  9488,  14825,   7236,   1225,    64;\n  8: 1, 116, 2601, 18896,  48025,  42516,  14161,  1856,   81;\n  9: 1, 132, 3537, 33168, 122025, 175716, 101577, 25152, 2673, 100;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+n+(12*n^3-2*n^2-11*n)*x+(16*n^4-4*n^3-40*n^2-5*n)*x^2-(16*n^4+76*n^3+94*n^2+49*n)*x^3+(4*n^3+7*n^2)*x^4)*P(n,x) = -(n^2+n+(12*n^3+30*n^2-7*n-28)*x+(16*n^4+156*n^3+80*n^2-231*n-58)*x^2+(112*n^4+356*n^3+42*n^2-499*n-378)*x^3-(76*n^3+289*n^2+374*n+190)*x^4+(8*n^2+22*n+14)*x^5)*P(n-1,x) + ((n+1)*x+(12*n^2+37*n+21)*x^2+(16*n^3+240*n^2+422*n+146)*x^3+(224*n^3+1016*n^2+1398*n+658)*x^4+(16*n^3+16*n^2-55*n-51)*x^5-(4*n^2+11*n+7)*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[(-(n^2 + n + (12*n^3 + 30*n^2 - 7*n - 28)*x + (16*n^4 + 156*n^3 + 80*n^2 - 231*n - 58)*x^2 + (112*n^4 + 356*n^3 + 42*n^2 - 499*n - 378)*x^3 - (76*n^3 + 289*n^2 + 374*n + 190)*x^4 + (8*n^2 + 22*n + 14)*x^5)*P[n - 1, x] + ((n + 1)*x + (12*n^2 + 37*n + 21)*x^2 + (16*n^3 + 240*n^2 + 422*n + 146)*x^3 + (224*n^3 + 1016*n^2 + 1398*n + 658)*x^4 + (16*n^3 + 16*n^2 - 55*n - 51)*x^5 - (4*n^2 + 11*n + 7)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (12*n^3 - 2*n^2 - 11*n)*x - (16*n^4 - 4*n^3 - 40*n^2 - 5*n)*x^2 + (16*n^4 + 76*n^3 + 94*n^2 + 49*n)*x^3 - (4*n^3 + 7*n^2)*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] = (-(n^2 + n + (12*n^3 + 30*n^2 - 7*n - 28)*x + (16*n^4 + 156*n^3 + 80*n^2 - 231*n - 58)*x^2 + (112*n^4 + 356*n^3 + 42*n^2 - 499*n - 378)*x^3 - (76*n^3 + 289*n^2 + 374*n + 190)*x^4 + (8*n^2 + 22*n + 14)*x^5)*v[i - 1] + ((n + 1)*x + (12*n^2 + 37*n + 21)*x^2 + (16*n^3 + 240*n^2 + 422*n + 146)*x^3 + (224*n^3 + 1016*n^2 + 1398*n + 658)*x^4 + (16*n^3 + 16*n^2 - 55*n - 51)*x^5 - (4*n^2 + 11*n + 7)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (12*n^3 - 2*n^2 - 11*n)*x - (16*n^4 - 4*n^3 - 40*n^2 - 5*n)*x^2 + (16*n^4 + 76*n^3 + 94*n^2 + 49*n)*x^3 - (4*n^3 + 7*n^2)*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-d8f9ad9ce0e039d0","steps":["(\u03b8+1)\u00b2"],"edges":["e-80bca760e085f131"],"text":"A081578 \u2192 (\u03b8+1)\u00b2"},"headline":"A081578 \u2192 (\u03b8+1)\u00b2","chain":null}