{"id":"n-ce9f6d305461a3cc","revision":0,"first_index":0,"fingerprint":"f6-53fd623c7116840c","name":"theta_plus_one_squared of Another version of Fibonacci-Pascal triangle A037027.","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"],["0","4"],["0","4","18"],["0","0","18","48"],["0","0","9","80","125"],["0","0","0","48","250","288"],["0","0","0","16","225","720","637"],["0","0","0","0","100","792","1862","1344"],["0","0","0","0","25","504","2499","4544","2754"],["0","0","0","0","0","180","1960","7104","10530","5500"],["0","0","0","0","0","36","980","6720","18873","23500","10769"],["0","0","0","0","0","0","294","4160","20736","47400","50820","20736"],["0","0","0","0","0","0","49","1728","15390","59400","113982","107136","39377"],["0","0","0","0","0","0","0","448","7938","51100","160204","264384","221052","73892"],["0","0","0","0","0","0","0","64","2835","31500","156695","411840","595218","447860","137250"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","66","214","586","1598","4098","10326","25274","60878","144146","337062","779018","1783262"],"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"],["0","4"]],"recurrence":{"denominator":{"coeffs":[["0","-4","14","-10","0"],["0","-10","-26","50","0"],["1","-7","-50","-75","25"],["2","11","20","-15","-50"],["1","8","26","40","25"]]},"terms":[{"coeff":{"coeffs":[["0","4","2","-18","-26","-10","0"],["0","12","53","79","53","15","0"],["0","13","87","240","286","145","25"],["0","6","49","167","269","195","50"],["0","1","9","34","66","65","25"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-35","-45","5","25"],["0","4","-27","-122","-65","0"],["0","5","19","5","-85","-50"],["0","4","27","72","65","0"],["0","1","8","26","40","25"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171896,"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-48c384a60492b25b"]},{"id":171897,"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-48c384a60492b25b"]},{"id":171898,"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-48c384a60492b25b"]}],"region_exact":{},"incoming_edges":[{"id":"e-48c384a60492b25b","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-ce9f6d305461a3cc","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":"e91575ac7fb3a6f88d80120ec68aefe653e8586980040c6c863a40dab14ddb6b","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-e7c079f81f452bf3","role":"sequence","name":"Another version of Fibonacci-Pascal triangle A037027.","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: 0, 4;\n  2: 0, 4, 18;\n  3: 0, 0, 18, 48;\n  4: 0, 0,  9, 80, 125;\n  5: 0, 0,  0, 48, 250, 288;\n  6: 0, 0,  0, 16, 225, 720,  637;\n  7: 0, 0,  0,  0, 100, 792, 1862, 1344;\n  8: 0, 0,  0,  0,  25, 504, 2499, 4544,  2754;\n  9: 0, 0,  0,  0,   0, 180, 1960, 7104, 10530, 5500;\n","recurrence_sentence":"The row polynomials satisfy (n^4+2*n^3+n^2+(8*n^4+11*n^3-7*n^2-10*n-4)*x+(26*n^4+20*n^3-50*n^2-26*n+14)*x^2+(40*n^4-15*n^3-75*n^2+50*n-10)*x^3+(25*n^4-50*n^3+25*n^2)*x^4)*P(n,x) = ((n^4+6*n^3+13*n^2+12*n+4)*x+(9*n^4+49*n^3+87*n^2+53*n+2)*x^2+(34*n^4+167*n^3+240*n^2+79*n-18)*x^3+(66*n^4+269*n^3+286*n^2+53*n-26)*x^4+(65*n^4+195*n^3+145*n^2+15*n-10)*x^5+(25*n^4+50*n^3+25*n^2)*x^6)*P(n-2,x) + ((n^4+4*n^3+5*n^2+4*n+2)*x+(8*n^4+27*n^3+19*n^2-27*n-35)*x^2+(26*n^4+72*n^3+5*n^2-122*n-45)*x^3+(40*n^4+65*n^3-85*n^2-65*n+5)*x^4+(25*n^4-50*n^2+25)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x + (9*n^4 + 49*n^3 + 87*n^2 + 53*n + 2)*x^2 + (34*n^4 + 167*n^3 + 240*n^2 + 79*n - 18)*x^3 + (66*n^4 + 269*n^3 + 286*n^2 + 53*n - 26)*x^4 + (65*n^4 + 195*n^3 + 145*n^2 + 15*n - 10)*x^5 + (25*n^4 + 50*n^3 + 25*n^2)*x^6)*P[n - 2, x] + ((n^4 + 4*n^3 + 5*n^2 + 4*n + 2)*x + (8*n^4 + 27*n^3 + 19*n^2 - 27*n - 35)*x^2 + (26*n^4 + 72*n^3 + 5*n^2 - 122*n - 45)*x^3 + (40*n^4 + 65*n^3 - 85*n^2 - 65*n + 5)*x^4 + (25*n^4 - 50*n^2 + 25)*x^5)*P[n - 1, x])/(n^4 + 2*n^3 + n^2 + (8*n^4 + 11*n^3 - 7*n^2 - 10*n - 4)*x + (26*n^4 + 20*n^3 - 50*n^2 - 26*n + 14)*x^2 + (40*n^4 - 15*n^3 - 75*n^2 + 50*n - 10)*x^3 + (25*n^4 - 50*n^3 + 25*n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x + (9*n^4 + 49*n^3 + 87*n^2 + 53*n + 2)*x^2 + (34*n^4 + 167*n^3 + 240*n^2 + 79*n - 18)*x^3 + (66*n^4 + 269*n^3 + 286*n^2 + 53*n - 26)*x^4 + (65*n^4 + 195*n^3 + 145*n^2 + 15*n - 10)*x^5 + (25*n^4 + 50*n^3 + 25*n^2)*x^6)*v[i - 2] + ((n^4 + 4*n^3 + 5*n^2 + 4*n + 2)*x + (8*n^4 + 27*n^3 + 19*n^2 - 27*n - 35)*x^2 + (26*n^4 + 72*n^3 + 5*n^2 - 122*n - 45)*x^3 + (40*n^4 + 65*n^3 - 85*n^2 - 65*n + 5)*x^4 + (25*n^4 - 50*n^2 + 25)*x^5)*v[i - 1])/(n^4 + 2*n^3 + n^2 + (8*n^4 + 11*n^3 - 7*n^2 - 10*n - 4)*x + (26*n^4 + 20*n^3 - 50*n^2 - 26*n + 14)*x^2 + (40*n^4 - 15*n^3 - 75*n^2 + 50*n - 10)*x^3 + (25*n^4 - 50*n^3 + 25*n^2)*x^4)); 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-e7c079f81f452bf3","steps":["(\u03b8+1)\u00b2"],"edges":["e-48c384a60492b25b"],"text":"A111006 \u2192 (\u03b8+1)\u00b2"},"headline":"A111006 \u2192 (\u03b8+1)\u00b2","chain":null}