{"id":"n-b82f6f873918cdd2","revision":1,"first_index":1,"fingerprint":"f6-d402ff27ba37b944","name":"theta_plus_one of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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"],["1","6"],["1","20","15"],["1","42","105","28"],["1","72","378","336","45"],["1","110","990","1848","825","66"],["1","156","2145","6864","6435","1716","91"],["1","210","4095","20020","32175","18018","3185","120"],["1","272","7140","49504","121550","116688","43316","5440","153"],["1","342","11628","108528","377910","554268","352716","93024","8721","190"],["1","420","17955","217056","1017450","2116296","2057510","930240","183141","13300","231"],["1","506","26565","403788","2451570","6864396","9464546","6537520","2206413","336490","19481","276"],["1","600","37950","708400","5407875","19612560","36402100","35659200","18386775","4807000","584430","27600","325"],["1","702","52650","1184040","11100375","50617710","121687020","160466400","117341055","46868250","9768330","968760","38025","378"],["1","812","71253","1900080","21460725","120180060","363271545","620470080","610775235","345972900","110165055","18729360","1543815","51156","435"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","36","176","832","3840","17408","77824","344064","1507328","6553600","28311552","121634816","520093696","2214592512"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["9","-3"],["-6","11"],["0","-12"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["9","51","12"],["-6","-25","-5"],["0","-12","-20"],["0","4","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-12","6","6"],["0","0","0","0"],["0","0","8","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164956,"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-1081279861cd2a9c"]},{"id":164957,"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-1081279861cd2a9c"]},{"id":164958,"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-1081279861cd2a9c"]},{"id":164959,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-1081279861cd2a9c"]}],"region_exact":{},"incoming_edges":[{"id":"e-1081279861cd2a9c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-b82f6f873918cdd2","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":"24cacb0339c8b8215f5f3519505bdfa5803ddf7f29516a9d3376d6694322fab3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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: 1,   6;\n   3: 1,  20,    15;\n   4: 1,  42,   105,     28;\n   5: 1,  72,   378,    336,     45;\n   6: 1, 110,   990,   1848,    825,     66;\n   7: 1, 156,  2145,   6864,   6435,   1716,     91;\n   8: 1, 210,  4095,  20020,  32175,  18018,   3185,   120;\n   9: 1, 272,  7140,  49504, 121550, 116688,  43316,  5440,  153;\n  10: 1, 342, 11628, 108528, 377910, 554268, 352716, 93024, 8721, 190;\n","recurrence_sentence":"The row polynomials satisfy -(6*n-9-(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = -(6*n-9-(4*n^3-12*n^2-25*n+51)*x-(12*n^3-20*n^2-5*n+12)*x^2)*P(n-1,x) - (12*x-(8*n^2+6)*x^2+(8*n^2-6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*P[n - 1, x] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*D[P[n - 1, x], x])/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*v[i - 1] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*deriv(v[i - 1], x))/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); 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-ba8de4e02acfb6ca","steps":["\u03b8+1"],"edges":["e-1081279861cd2a9c"],"text":"A091042 \u2192 \u03b8+1"},"headline":"A091042 \u2192 \u03b8+1","chain":null}
