{"id":"n-a618dbaa2ab6e2bf","revision":0,"first_index":1,"fingerprint":"f6-48e60cd64bbf3ade","name":"theta_plus_one_squared of basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","4"],["6","20","9"],["16","80","72","16"],["44","288","387","176","25"],["120","976","1746","1200","350","36"],["328","3184","7110","6544","2900","612","49"],["896","10112","27072","31232","18600","5976","980","64"],["2448","31488","98280","136320","102025","44136","11025","1472","81"],["6688","96576","344304","557952","503050","273924","92218","18752","2106","100"],["18272","292672","1173312","2175360","2294950","1507428","639401","175296","29970","2900","121"],["49920","878336","3911040","8164608","9866800","7591824","3887856","1343680","309744","45600","3872","144"],["136384","2614784","12803472","29718784","40479700","35711424","21439607","8951104","2601882","516400","66671","5040","169"],["372608","7731456","41288544","105481472","159885800","159105456","109595458","53670336","18852588","4718600","821106","94320","6422","196"],["1017984","22728448","131463072","366564096","611981400","678160080","527237550","296390720","122164200","36957000","8109904","1255248","129792","8036","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","35","184","920","4428","20727","94932","427275","1895670","8309682","36053424","155045421","661624362","2804167755"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["576","-960","64","128","8","-2"],["0","864","-504","-120","26","3"],["0","0","312","-44","-29","0"],["0","0","0","36","2","-1"],["0","0","0","0","1","0"]]},"terms":[{"coeff":{"coeffs":[["1152","384","-352","-128","16","8","0"],["0","1728","1104","-80","-144","-22","0"],["0","0","624","440","30","-23","-3"],["0","0","0","72","48","4","-1"],["0","0","0","0","2","1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1152","384","-352","-128","16","8","0"],["0","1728","768","-352","-188","-18","0"],["0","0","624","368","-18","-27","-2"],["0","0","0","72","44","2","-1"],["0","0","0","0","2","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175696,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f55953bd01542cd5"]},{"id":175697,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f55953bd01542cd5"]},{"id":175698,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f55953bd01542cd5"]}],"region_exact":{},"incoming_edges":[{"id":"e-f55953bd01542cd5","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-a618dbaa2ab6e2bf","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":"b445ab82ff119576360779d887bcb741e813c286e2f161d695ffcafa6472c311","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c63944481deaae45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    2,     4;\n   3:    6,    20,      9;\n   4:   16,    80,     72,     16;\n   5:   44,   288,    387,    176,     25;\n   6:  120,   976,   1746,   1200,    350,     36;\n   7:  328,  3184,   7110,   6544,   2900,    612,    49;\n   8:  896, 10112,  27072,  31232,  18600,   5976,   980,    64;\n   9: 2448, 31488,  98280, 136320, 102025,  44136, 11025,  1472,   81;\n  10: 6688, 96576, 344304, 557952, 503050, 273924, 92218, 18752, 2106, 100;\n","recurrence_sentence":"The row polynomials satisfy (576+(864*n-960)*x+(312*n^2-504*n+64)*x^2+(36*n^3-44*n^2-120*n+128)*x^3+(n^4+2*n^3-29*n^2+26*n+8)*x^4-(n^3-3*n+2)*x^5)*P(n,x) = (1152+(1728*n+384)*x+(624*n^2+1104*n-352)*x^2+(72*n^3+440*n^2-80*n-128)*x^3+(2*n^4+48*n^3+30*n^2-144*n+16)*x^4+(n^4+4*n^3-23*n^2-22*n+8)*x^5-(n^3+3*n^2)*x^6)*P(n-2,x) + (1152+(1728*n+384)*x+(624*n^2+768*n-352)*x^2+(72*n^3+368*n^2-352*n-128)*x^3+(2*n^4+44*n^3-18*n^2-188*n+16)*x^4+(n^4+2*n^3-27*n^2-18*n+8)*x^5-(n^3+2*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((1152 + (1728*n + 384)*x + (624*n^2 + 1104*n - 352)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (n^4 + 4*n^3 - 23*n^2 - 22*n + 8)*x^5 - (n^3 + 3*n^2)*x^6)*P[n - 2, x] + (1152 + (1728*n + 384)*x + (624*n^2 + 768*n - 352)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (n^4 + 2*n^3 - 27*n^2 - 18*n + 8)*x^5 - (n^3 + 2*n^2)*x^6)*P[n - 1, x])/(576 + (864*n - 960)*x + (312*n^2 - 504*n + 64)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (n^3 - 3*n + 2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((1152 + (1728*n + 384)*x + (624*n^2 + 1104*n - 352)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (n^4 + 4*n^3 - 23*n^2 - 22*n + 8)*x^5 - (n^3 + 3*n^2)*x^6)*v[i - 2] + (1152 + (1728*n + 384)*x + (624*n^2 + 768*n - 352)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (n^4 + 2*n^3 - 27*n^2 - 18*n + 8)*x^5 - (n^3 + 2*n^2)*x^6)*v[i - 1])/(576 + (864*n - 960)*x + (312*n^2 - 504*n + 64)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (n^3 - 3*n + 2)*x^5)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-3a20d12bade77e7f","e-f55953bd01542cd5"],"text":"A063967 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A063967 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}