{"id":"n-8d4eedf9d66a931a","revision":1,"first_index":1,"fingerprint":"f6-bd81fa9c8482d6ca","name":"theta_plus_one_squared 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.","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"],["1","4"],["2","12","9"],["3","28","45","16"],["5","60","144","112","25"],["8","120","387","464","225","36"],["13","232","936","1520","1150","396","49"],["21","436","2115","4336","4475","2412","637","64"],["34","804","4554","11280","14775","10908","4508","960","81"],["55","1460","9459","27472","43650","41040","23275","7744","1377","100"],["89","2620","19107","63664","118975","136080","98490","44992","12474","1900","121"],["144","4656","37746","141920","305025","411300","362845","211776","80595","19100","2541","144"],["233","8208","73224","306656","745225","1157940","1206380","859328","418041","135900","28072","3312","169"],["377","14372","139914","645856","1751150","3081600","3705135","3120704","1854252","770600","218163","39888","4225","196"],["610","25020","263943","1331424","3984675","7834320","10682000","10395072","7309521","3711800","1343100","336240","55094","5292","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","23","92","346","1240","4296","14496","47904","155632","498512","1577792","4942688","15346432","47278336"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["25","-85","34","26","1","-1"],["0","75","-110","-18","14","1"],["0","0","60","-25","-13","1"],["0","0","0","15","0","-1"],["0","0","0","0","1","0"]]},"terms":[{"coeff":{"coeffs":[["25","15","-26","-18","1","3","0"],["0","75","85","-13","-31","-8","0"],["0","0","60","80","13","-9","-2"],["0","0","0","15","19","3","-1"],["0","0","0","0","1","1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["25","15","-26","-18","1","3","0"],["0","75","50","-68","-48","-5","0"],["0","0","60","65","-6","-12","-1"],["0","0","0","15","17","1","-1"],["0","0","0","0","1","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163612,"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-aa7e9b3181553556"]},{"id":163613,"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-aa7e9b3181553556"]},{"id":163614,"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-aa7e9b3181553556"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa7e9b3181553556","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-8d4eedf9d66a931a","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":"96163c8d326bf85052d013c567317cf600fc44da2410e6548c396b69fdd4158d","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-c72653d9ab0ee8be","role":"sequence","name":"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.","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:  1,    4;\n   3:  2,   12,    9;\n   4:  3,   28,   45,    16;\n   5:  5,   60,  144,   112,    25;\n   6:  8,  120,  387,   464,   225,    36;\n   7: 13,  232,  936,  1520,  1150,   396,    49;\n   8: 21,  436, 2115,  4336,  4475,  2412,   637,   64;\n   9: 34,  804, 4554, 11280, 14775, 10908,  4508,  960,   81;\n  10: 55, 1460, 9459, 27472, 43650, 41040, 23275, 7744, 1377, 100;\n","recurrence_sentence":"The row polynomials satisfy (25+(75*n-85)*x+(60*n^2-110*n+34)*x^2+(15*n^3-25*n^2-18*n+26)*x^3+(n^4-13*n^2+14*n+1)*x^4-(n^3-n^2-n+1)*x^5)*P(n,x) = (25+(75*n+15)*x+(60*n^2+85*n-26)*x^2+(15*n^3+80*n^2-13*n-18)*x^3+(n^4+19*n^3+13*n^2-31*n+1)*x^4+(n^4+3*n^3-9*n^2-8*n+3)*x^5-(n^3+2*n^2)*x^6)*P(n-2,x) + (25+(75*n+15)*x+(60*n^2+50*n-26)*x^2+(15*n^3+65*n^2-68*n-18)*x^3+(n^4+17*n^3-6*n^2-48*n+1)*x^4+(n^4+n^3-12*n^2-5*n+3)*x^5-(n^3+n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((25 + (75*n + 15)*x + (60*n^2 + 85*n - 26)*x^2 + (15*n^3 + 80*n^2 - 13*n - 18)*x^3 + (n^4 + 19*n^3 + 13*n^2 - 31*n + 1)*x^4 + (n^4 + 3*n^3 - 9*n^2 - 8*n + 3)*x^5 - (n^3 + 2*n^2)*x^6)*P[n - 2, x] + (25 + (75*n + 15)*x + (60*n^2 + 50*n - 26)*x^2 + (15*n^3 + 65*n^2 - 68*n - 18)*x^3 + (n^4 + 17*n^3 - 6*n^2 - 48*n + 1)*x^4 + (n^4 + n^3 - 12*n^2 - 5*n + 3)*x^5 - (n^3 + n^2)*x^6)*P[n - 1, x])/(25 + (75*n - 85)*x + (60*n^2 - 110*n + 34)*x^2 + (15*n^3 - 25*n^2 - 18*n + 26)*x^3 + (n^4 - 13*n^2 + 14*n + 1)*x^4 - (n^3 - n^2 - n + 1)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((25 + (75*n + 15)*x + (60*n^2 + 85*n - 26)*x^2 + (15*n^3 + 80*n^2 - 13*n - 18)*x^3 + (n^4 + 19*n^3 + 13*n^2 - 31*n + 1)*x^4 + (n^4 + 3*n^3 - 9*n^2 - 8*n + 3)*x^5 - (n^3 + 2*n^2)*x^6)*v[i - 2] + (25 + (75*n + 15)*x + (60*n^2 + 50*n - 26)*x^2 + (15*n^3 + 65*n^2 - 68*n - 18)*x^3 + (n^4 + 17*n^3 - 6*n^2 - 48*n + 1)*x^4 + (n^4 + n^3 - 12*n^2 - 5*n + 3)*x^5 - (n^3 + n^2)*x^6)*v[i - 1])/(25 + (75*n - 85)*x + (60*n^2 - 110*n + 34)*x^2 + (15*n^3 - 25*n^2 - 18*n + 26)*x^3 + (n^4 - 13*n^2 + 14*n + 1)*x^4 - (n^3 - n^2 - n + 1)*x^5)); 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-c72653d9ab0ee8be","steps":["(\u03b8+1)\u00b2"],"edges":["e-aa7e9b3181553556"],"text":"A063967 \u2192 (\u03b8+1)\u00b2"},"headline":"A063967 \u2192 (\u03b8+1)\u00b2","chain":null}