{"id":"n-73c8436fb109f7ec","revision":0,"first_index":0,"fingerprint":"f6-3dfc77d38090b9d0","name":"theta_plus_one of gamma_polynomial of Pascal-(1,7,1) array.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","14"],["1","28"],["1","42","147"],["1","56","441"],["1","70","882","1372"],["1","84","1470","5488"],["1","98","2205","13720","12005"],["1","112","3087","27440","60025"],["1","126","4116","48020","180075","100842"],["1","140","5292","76832","420175","605052"],["1","154","6615","115248","840350","2117682","823543"],["1","168","8085","164640","1512630","5647152","5764801"],["1","182","9702","226380","2521050","12706092","23059204","6588344"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","15","29","190","498","2325","7043","28029","90665","333180","1107492","3903593","13097477","45110955"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["1","0"],["0","7"]]},"terms":[{"coeff":{"coeffs":[["0","-28","-196"],["1","28","98"],["0","7","98"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-42","-392"],["0","0","-7","-196"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174324,"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-0b03d17618abd12e"]},{"id":174325,"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-0b03d17618abd12e"]},{"id":174326,"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-0b03d17618abd12e"]}],"region_exact":{},"incoming_edges":[{"id":"e-0b03d17618abd12e","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-73c8436fb109f7ec","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":"ff8fdc47e30c506eafe5805abd7e1a87afe066a0308ffd0f1917362826ac3a27","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-695d6cb541611b1e","role":"sequence","name":"gamma_polynomial of Pascal-(1,7,1) 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;\n  2: 1,  14;\n  3: 1,  28;\n  4: 1,  42,  147;\n  5: 1,  56,  441;\n  6: 1,  70,  882,  1372;\n  7: 1,  84, 1470,  5488;\n  8: 1,  98, 2205, 13720, 12005;\n  9: 1, 112, 3087, 27440, 60025;\n","recurrence_sentence":"The row polynomials satisfy (n+7*n^2*x)*P(n,x) = (n+(7*n^2+28*n-28)*x+(98*n^2+98*n-196)*x^2)*P(n-1,x) - (x+(7*n+42)*x^2+(196*n+392)*x^3)*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 + (7*n^2 + 28*n - 28)*x + (98*n^2 + 98*n - 196)*x^2)*P[n - 1, x] - (x + (7*n + 42)*x^2 + (196*n + 392)*x^3)*D[P[n - 1, x], x])/(n + 7*n^2*x)]];\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 + (7*n^2 + 28*n - 28)*x + (98*n^2 + 98*n - 196)*x^2)*v[i - 1] - (x + (7*n + 42)*x^2 + (196*n + 392)*x^3)*deriv(v[i - 1], x))/(n + 7*n^2*x)); 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-1e2f1b7f566b371c","steps":["\u03b3","\u03b8+1"],"edges":["e-c807fb938e5557d1","e-0b03d17618abd12e"],"text":"A081582 \u2192 \u03b3 \u2192 \u03b8+1"},"headline":"A081582 \u2192 \u03b3 \u2192 \u03b8+1","chain":null}