{"id":"n-a2f64418470a1afb","revision":0,"first_index":0,"fingerprint":"f6-f7ea65e9d1891b99","name":"theta_plus_one of derivative of Even row Pascal-square read by antidiagonals.","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":[[],["2"],["4","12"],["6","40","36"],["8","84","180","80"],["10","144","504","560","150"],["12","220","1080","2016","1400","252"],["14","312","1980","5280","6300","3024","392"],["16","420","3276","11440","19800","16632","5880","576"],["18","544","5040","21840","50050","61776","38808","10560","810"],["20","684","7344","38080","109200","180180","168168","82368","17820","1100"],["22","840","10260","62016","214200","445536","560560","411840","162162","28600","1452"],["24","1012","13860","95760","387600","976752","1559376","1555840","926640","300300","44044","1872"],["26","1200","18216","141680","658350","1953504","3798480","4837248","3938220","1944800","528528","65520","2366"],["28","1404","23400","202400","1062600","3634092","8356656","13023360","13604760","9237800","3850704","891072","94640","2940"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","16","82","352","1368","4980","17302","58040","189446","604964","1897488","5863080","17888138","53985856"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["0","2"],["2","-12"],["-6","26"],["4","-24"],["0","8"]]},"terms":[{"coeff":{"coeffs":[["0","0","8"],["0","-6","6"],["-2","-18","-10"],["4","8","-32"],["0","8","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","4"],["0","0","1","4"],["0","-2","-8","0"],["0","0","-4","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174162,"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-e76ae0102f0458a0"]},{"id":174163,"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-e76ae0102f0458a0"]},{"id":174164,"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-e76ae0102f0458a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-e76ae0102f0458a0","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-a2f64418470a1afb","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":"ccb91016d714bd3526eae7742f4044ad89feeb750165816ba4c277208e1b6735","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-69a7d6f31c3b5900","role":"sequence","name":"derivative of Even row Pascal-square read by antidiagonals.","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:  0;\n  1:  2;\n  2:  4,  12;\n  3:  6,  40,   36;\n  4:  8,  84,  180,    80;\n  5: 10, 144,  504,   560,   150;\n  6: 12, 220, 1080,  2016,  1400,   252;\n  7: 14, 312, 1980,  5280,  6300,  3024,   392;\n  8: 16, 420, 3276, 11440, 19800, 16632,  5880,   576;\n  9: 18, 544, 5040, 21840, 50050, 61776, 38808, 10560, 810;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2+2*n+(8*n^4-24*n^3+26*n^2-12*n+2)*x)*P(n,x) = (4*n^3-2*n^2+(8*n^4+8*n^3-18*n^2-6*n)*x+(24*n^4-32*n^3-10*n^2+6*n+8)*x^2)*P(n-1,x) - (2*n^2*x+(4*n^3+8*n^2-n-1)*x^2+(16*n^3-4*n-4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 + (8*n^4 + 8*n^3 - 18*n^2 - 6*n)*x + (24*n^4 - 32*n^3 - 10*n^2 + 6*n + 8)*x^2)*P[n - 1, x] - (2*n^2*x + (4*n^3 + 8*n^2 - n - 1)*x^2 + (16*n^3 - 4*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n + (8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^3 - 2*n^2 + (8*n^4 + 8*n^3 - 18*n^2 - 6*n)*x + (24*n^4 - 32*n^3 - 10*n^2 + 6*n + 8)*x^2)*v[i - 1] - (2*n^2*x + (4*n^3 + 8*n^2 - n - 1)*x^2 + (16*n^3 - 4*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 + 2*n + (8*n^4 - 24*n^3 + 26*n^2 - 12*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-79ebafb266da3b41","steps":["D","\u03b8+1"],"edges":["e-032b739c94c6fe41","e-e76ae0102f0458a0"],"text":"A172431 \u2192 D \u2192 \u03b8+1"},"headline":"A172431 \u2192 D \u2192 \u03b8+1","chain":null}