{"id":"n-8fac5e5e83554770","revision":0,"first_index":0,"fingerprint":"f6-831f2354c620f023","name":"theta_plus_one of basis_map_A007318 of Riordan array (1, 2+2x).","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"],["2","4"],["6","20","12"],["16","80","96","32"],["44","288","516","352","80"],["120","976","2328","2400","1120","192"],["328","3184","9480","13088","9280","3264","448"],["896","10112","36096","62464","59520","31872","8960","1024"],["2448","31488","131040","272640","326480","235392","100800","23552","2304"],["6688","96576","459072","1115904","1609760","1460928","843136","300032","59904","5120"],["18272","292672","1564416","4350720","7343840","8039616","5845952","2804736","852480","148480","11264"],["49920","878336","5214720","16329216","31573760","40489728","35546112","21498880","8810496","2334720","360448","24576"],["136384","2614784","17071296","59437568","129535040","190460928","196019264","143217664","74009088","26439680","6206464","860160","53248"],["372608","7731456","55051392","210962944","511634560","848562432","1002015616","858725376","536251392","241592320","76437504","16097280","2023424","114688"],["1017984","22728448","175284096","733128192","1958340480","3616853760","4820457600","4742251520","3474892800","1892198400","754958336","214228992","40894464","4702208","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","38","224","1280","7136","39072","210944","1126144","5957120","31272448","163110912","846061568","4367572992","22452183040"],"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","0"],["12","16","4"],["0","6","5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["48","100","68","16"],["-24","10","48","20"],["0","-18","-6","4"],["0","0","-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["36","108","116","52","8"],["0","36","78","52","10"],["0","0","6","8","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177727,"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-cde353f468fbd8c3"]},{"id":177728,"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-cde353f468fbd8c3"]},{"id":177729,"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-cde353f468fbd8c3"]}],"region_exact":{},"incoming_edges":[{"id":"e-cde353f468fbd8c3","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-8fac5e5e83554770","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":"7568d134310dc1e6449ac70b077e372f26718b4b25132f90623c9a913633b814","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-002f74523f94b1c9","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 2+2x).","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-ccdf8a563fb938b7","kind":"fingerprint_related","node_a_id":"n-8fac5e5e83554770","node_b_id":"n-9be176da304a3877","fingerprint":"f6-831f2354c620f023","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-831f2354c620f023"},"other_id":"n-9be176da304a3877","other":{"id":"n-9be176da304a3877","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","first_index":1,"fingerprint":"f6-831f2354c620f023","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","20","12"]],"card_rows":[["1"],["2","4"],["6","20","12"],["16","80","96","32"]],"row_sums_preview":["1","6","38","224","1280","7136","39072","210944"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A013609","\u03b8+1"],"edges":["e-081205c1b1ff787d","e-6f98647f5577d726"],"text":"A063967 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A063967 \u2192 basis A013609 \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:    2,     4;\n  2:    6,    20,     12;\n  3:   16,    80,     96,      32;\n  4:   44,   288,    516,     352,      80;\n  5:  120,   976,   2328,    2400,    1120,     192;\n  6:  328,  3184,   9480,   13088,    9280,    3264,    448;\n  7:  896, 10112,  36096,   62464,   59520,   31872,   8960,   1024;\n  8: 2448, 31488, 131040,  272640,  326480,  235392, 100800,  23552,  2304;\n  9: 6688, 96576, 459072, 1115904, 1609760, 1460928, 843136, 300032, 59904, 5120;\n","recurrence_sentence":"The row polynomials satisfy (12*n+(6*n^2+16*n)*x+(n^3+5*n^2+4*n)*x^2)*P(n,x) = -(24*n-48+(18*n^2-10*n-100)*x+(2*n^3+6*n^2-48*n-68)*x^2-(4*n^2+20*n+16)*x^3)*P(n-1,x) + (36+(36*n+108)*x+(6*n^2+78*n+116)*x^2+(8*n^2+52*n+52)*x^3+(2*n^2+10*n+8)*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[(-(24*n - 48 + (18*n^2 - 10*n - 100)*x + (2*n^3 + 6*n^2 - 48*n - 68)*x^2 - (4*n^2 + 20*n + 16)*x^3)*P[n - 1, x] + (36 + (36*n + 108)*x + (6*n^2 + 78*n + 116)*x^2 + (8*n^2 + 52*n + 52)*x^3 + (2*n^2 + 10*n + 8)*x^4)*D[P[n - 1, x], x])/(12*n + (6*n^2 + 16*n)*x + (n^3 + 5*n^2 + 4*n)*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] = (-(24*n - 48 + (18*n^2 - 10*n - 100)*x + (2*n^3 + 6*n^2 - 48*n - 68)*x^2 - (4*n^2 + 20*n + 16)*x^3)*v[i - 1] + (36 + (36*n + 108)*x + (6*n^2 + 78*n + 116)*x^2 + (8*n^2 + 52*n + 52)*x^3 + (2*n^2 + 10*n + 8)*x^4)*deriv(v[i - 1], x))/(12*n + (6*n^2 + 16*n)*x + (n^3 + 5*n^2 + 4*n)*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-f5a001953ee28e8f","steps":["basis A007318","\u03b8+1"],"edges":["e-23c3ede392db8800","e-cde353f468fbd8c3"],"text":"A099040 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A099040 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}