{"id":"n-4a58db838ece5aa9","revision":0,"first_index":0,"fingerprint":"f6-b9193847b07222ca","name":"theta_plus_one_squared of gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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"],["2"],["4","8"],["8","32"],["16","96","36"],["32","256","216"],["64","640","864","128"],["128","1536","2880","1024"],["256","3584","8640","5120","400"],["512","8192","24192","20480","4000"],["1024","18432","64512","71680","24000","1152"],["2048","40960","165888","229376","112000","13824"],["4096","90112","414720","688128","448000","96768","3136"],["8192","196608","1013760","1966080","1612800","516096","43904"],["16384","425984","2433024","5406720","5376000","2322432","351232","8192"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","12","40","148","504","1696","5568","18000","57376","180800","564096","1744960","5357440","16339968"],"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"],["4","-4","0"],["4","0","0"],["0","6","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-32","-8","-8"],["8","-8","-28","-8"],["8","32","16","6"],["0","12","22","8"],["0","0","2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-24","-24","-16"],["0","-8","-40","-64","-32"],["0","0","-12","-34","-20"],["0","0","0","-2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173117,"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-a03a7531210fa543"]},{"id":173118,"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-a03a7531210fa543"]},{"id":173119,"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-a03a7531210fa543"]}],"region_exact":{},"incoming_edges":[{"id":"e-a03a7531210fa543","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-4a58db838ece5aa9","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":"9838e43f0fffadb7ad10e3a4daf4da95b4d5f50e11e381d173c84c8d0d8406f1","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-a5662819bbc56ccb","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:   2;\n  2:   4,    8;\n  3:   8,   32;\n  4:  16,   96,    36;\n  5:  32,  256,   216;\n  6:  64,  640,   864,   128;\n  7: 128, 1536,  2880,  1024;\n  8: 256, 3584,  8640,  5120,  400;\n  9: 512, 8192, 24192, 20480, 4000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n+(6*n^3-4*n)*x+(n^4+n^3)*x^2)*P(n,x) = (8*n^2+8*n+(12*n^3+32*n^2-8*n-32)*x+(2*n^4+22*n^3+16*n^2-28*n-8)*x^2+(2*n^4+8*n^3+6*n^2-8*n-8)*x^3)*P(n-1,x) - ((8*n+8)*x+(12*n^2+40*n+24)*x^2+(2*n^3+34*n^2+64*n+24)*x^3+(4*n^3+20*n^2+32*n+16)*x^4)*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[((8*n^2 + 8*n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (2*n^4 + 22*n^3 + 16*n^2 - 28*n - 8)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*P[n - 1, x] - ((8*n + 8)*x + (12*n^2 + 40*n + 24)*x^2 + (2*n^3 + 34*n^2 + 64*n + 24)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 4*n + (6*n^3 - 4*n)*x + (n^4 + n^3)*x^2)]];\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] = ((8*n^2 + 8*n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (2*n^4 + 22*n^3 + 16*n^2 - 28*n - 8)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*v[i - 1] - ((8*n + 8)*x + (12*n^2 + 40*n + 24)*x^2 + (2*n^3 + 34*n^2 + 64*n + 24)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 4*n + (6*n^3 - 4*n)*x + (n^4 + n^3)*x^2)); 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-1bffaa6198344f93","steps":["\u03b3","(\u03b8+1)\u00b2"],"edges":["e-babfdb0b6fa40ea8","e-a03a7531210fa543"],"text":"A059473 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2"},"headline":"A059473 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2","chain":null}