{"id":"n-3503b708139a235b","revision":0,"first_index":1,"fingerprint":"f6-110d29bcfc73cc03","name":"theta_plus_one of Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","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"],["0","2"],["0","4","3"],["0","10","15","4"],["0","32","69","36","5"],["0","130","348","260","70","6"],["0","652","2007","1880","725","120","7"],["0","3914","13287","14536","7075","1680","189","8"],["0","27400","100125","122724","70420","21210","3430","280","9"],["0","219202","849216","1136264","737660","263634","54292","6384","396","10"],["0","1972820","8019963","11513136","8233570","3351852","823053","123648","11070","540","11"],["0","19728202","83535879","127111404","98321750","44377950","12475449","2243304","257310","18150","715","12"],["0","217010224","951823221","1521586660","1256414555","617002980","193249133","39940824","5500737","498300","28435","924","13"],["0","2604122690","11780322540","19651498764","17153833290","9040578756","3094845600","715410520","113612598","12390180","909920","42900","1170","14"],["0","33853594972","157408820943","272569039496","249641064855","139748666772","51556190686","13073423792","2329404363","293719140","26026858","1582152","62699","1456","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","29","142","814","5391","40689","345598","3267058","34049663","388070125","4803056006","64167568942","920501598199"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","2"],["0","4","3"],["0","10","15","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-20"],["9"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["20","-8"],["-13","2"],["2","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["3","4","-1"],["2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177210,"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-8f29e1cc0cf6ac20"]},{"id":177211,"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-8f29e1cc0cf6ac20"]},{"id":177212,"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-8f29e1cc0cf6ac20"]}],"region_exact":{},"incoming_edges":[{"id":"e-8f29e1cc0cf6ac20","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-3503b708139a235b","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":"0d921b0219168fcd28993feee9bb98a666ba6516908e2a897282234426895921","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-cd3918419c5686a8","role":"sequence","name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","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: 0,      2;\n   3: 0,      4,      3;\n   4: 0,     10,     15,       4;\n   5: 0,     32,     69,      36,      5;\n   6: 0,    130,    348,     260,     70,      6;\n   7: 0,    652,   2007,    1880,    725,    120,     7;\n   8: 0,   3914,  13287,   14536,   7075,   1680,   189,    8;\n   9: 0,  27400, 100125,  122724,  70420,  21210,  3430,  280,   9;\n  10: 0, 219202, 849216, 1136264, 737660, 263634, 54292, 6384, 396, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(2*n-8)*x)*P(n-3,x) - (n^2-2*n-3+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 4*x + 3*x^2, P(4,x) = 10*x + 15*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[3, x_] := 4*x + 3*x^2;\nP[4, x_] := 10*x + 15*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*P[n - 3, x] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 4*x + 3*x^2; v[4] = 10*x + 15*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 9*n + 20)*v[i - 4] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*v[i - 3] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 2*x)*v[i - 1]); 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-cd3918419c5686a8","steps":["\u03b8+1"],"edges":["e-8f29e1cc0cf6ac20"],"text":"A269951 \u2192 \u03b8+1"},"headline":"A269951 \u2192 \u03b8+1","chain":null}