{"id":"n-5cf891f602fc8484","revision":0,"first_index":0,"fingerprint":"f6-fbfce332e6b0be6b","name":"theta_plus_one of Second-order Eulerian triangle T(n,k), 1 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","4"],["1","16","18"],["1","44","174","96"],["1","104","984","1776","600"],["1","228","4356","17600","18540","4320"],["1","480","16830","128480","290700","203904","35280"],["1","988","59850","783200","3220100","4711824","2387952","322560"],["1","2008","201780","4250000","28827500","74640384","77184072","29871360","3265920"],["1","4052","656544","21304640","223825000","932144304","1672334328","1297495296","399051360","36288000"],["1","8144","2085114","100975616","1571848600","9890305824","28018865616","37137311616","22567330944","5683564800","439084800"],["1","16332","6511878","459505504","10257413880","93293764704","392289791488","809443464192","829798112016","407883018240","86104909440","5748019200"],["1","32712","20099088","2029037104","63349088880","805940521344","4814040324688","14545509138432","22948789260816","18830796618240","7677223804800","1384033029120","80951270400"],["1","65476","61523964","8759319232","375080261140","6508059074496","53443825045744","226399280528896","517449361181328","647283751398720","436585358613888","150610848215040","23542903238400","1220496076800"],["1","131008","187223670","37170107680","2149130031700","49850665967808","548739693819184","3154924702524160","10005727253127120","17971712206905600","18330669348986016","10380279038635008","3079807530059520","422749277798400","19615115520000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","315","3465","45045","675675","11486475","218243025","4583103525","105411381075","2635284526875","71152682225625","2063427784543125","63966261320836875"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","2"],["0","0","2"],["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","1"],["0","0","-2","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184499,"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-8e7778f4e8f9a5ee"]},{"id":184500,"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-8e7778f4e8f9a5ee"]},{"id":184501,"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-8e7778f4e8f9a5ee"]},{"id":184502,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-8e7778f4e8f9a5ee"]}],"region_exact":{},"incoming_edges":[{"id":"e-8e7778f4e8f9a5ee","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-5cf891f602fc8484","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":"1f31d4e755558ade561bcc42ca74c937da2fc95a58d173db67645f69480ec618","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d4c21302c908a2cf","role":"sequence","name":"Second-order Eulerian triangle T(n,k), 1 <= k <= n.","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":[{"id":"l-1ea4ba887c17eb05","kind":"fingerprint_related","node_a_id":"n-00fabd7673b77000","node_b_id":"n-5cf891f602fc8484","fingerprint":"f6-fbfce332e6b0be6b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fbfce332e6b0be6b"},"other_id":"n-00fabd7673b77000","other":{"id":"n-00fabd7673b77000","name":"derivative of E2(n, k), the second-order Eulerian numbers with E2(0, k) = \u03b4_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-fbfce332e6b0be6b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","4"]],"card_rows":[["1"],["1","4"],["1","16","18"],["1","44","174","96"]],"row_sums_preview":["0","1","5","35","315","3465","45045","675675"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e14e37227fef094e","steps":["reverse","D"],"edges":["e-c8375d3b3fe52ed3","e-7b753f6ebcf5fd96"],"text":"A163936 \u2192 reverse \u2192 D"},"headline":"A163936 \u2192 reverse \u2192 D","chain":null}},{"id":"l-b718a7661df8b9a6","kind":"fingerprint_related","node_a_id":"n-8e982d8b673fb1df","node_b_id":"n-5cf891f602fc8484","fingerprint":"f6-fbfce332e6b0be6b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fbfce332e6b0be6b"},"other_id":"n-8e982d8b673fb1df","other":{"id":"n-8e982d8b673fb1df","name":"theta_plus_one of Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","first_index":1,"fingerprint":"f6-fbfce332e6b0be6b","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","16","18"]],"row_sums_preview":["1","1","5","35","315","3465","45045","675675"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e2f0f447a502f269","steps":["\u03b8+1"],"edges":["e-a2c97f659f2ce93d"],"text":"A201637 \u2192 \u03b8+1"},"headline":"A201637 \u2192 \u03b8+1","chain":null}}],"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,    4;\n  2: 1,   16,     18;\n  3: 1,   44,    174,       96;\n  4: 1,  104,    984,     1776,       600;\n  5: 1,  228,   4356,    17600,     18540,      4320;\n  6: 1,  480,  16830,   128480,    290700,    203904,      35280;\n  7: 1,  988,  59850,   783200,   3220100,   4711824,    2387952,     322560;\n  8: 1, 2008, 201780,  4250000,  28827500,  74640384,   77184072,   29871360,   3265920;\n  9: 1, 4052, 656544, 21304640, 223825000, 932144304, 1672334328, 1297495296, 399051360, 36288000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-2*n-2)*x^2*P(n-2,x) - ((2*n+1)*x^2-(2*n+1)*x^3)*d/dx P(n-2,x) + (1+4*n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 2*n - 2)*x^2*P[n - 2, x] - ((2*n + 1)*x^2 - (2*n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 4*n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 2*n - 2)*x^2*v[i - 2] - ((2*n + 1)*x^2 - (2*n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 4*n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], 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-d4c21302c908a2cf","steps":["\u03b8+1"],"edges":["e-8e7778f4e8f9a5ee"],"text":"A008517 \u2192 \u03b8+1"},"headline":"A008517 \u2192 \u03b8+1","chain":null}