{"id":"n-f56f01f99b47548b","revision":0,"first_index":0,"fingerprint":"f6-3ac78d4b619e6815","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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","20","9"],["1","84","126","16"],["1","340","1323","480","25"],["1","1364","12672","10032","1375","36"],["1","5460","117117","183040","50050","3276","49"],["1","21844","1066338","3136848","1537250","190008","6860","64"],["1","87380","9646191","52085280","43332575","9053928","594762","13056","81"],["1","349524","87012324","850513264","1164697625","388340316","41466740","1612416","23085","100"],["1","1398100","783897345","13762900800","30446367600","15657415956","2560444579","157355264","3910599","38500","121"],["1","5592404","7058221830","221600008080","782663722500","607509743760","146773267200","13414991040","515911275","8677900","61226","144"],["1","22369620","63536579379","3558148079200","19912843075125","22997386535760","8018778355140","1050263285760","58767161310","1504717500","17908605","93600","169"],["1","89478484","571879546056","57043323186096","503380683251875","856580409315540","424222137742200","77690356711680","6089379537150","223023801000","3987649380","34791120","138411","196"],["1","357913940","5147117241093","913709845726080","12673647273775150","31561762919242140","21952785862047285","5526268887006720","591566600222370","29820132615000","752364374190","9755553600","64222704","198940","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","227","2169","25480","358993","5959213","114813254","2534015395","63373728865","1779550197360","55661268161169","1925805305149189","73255471180139438"],"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"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","9","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","17"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","4"]]},"deriv_order":3,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182891,"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-3933e870d6e631c7"]},{"id":182892,"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-3933e870d6e631c7"]},{"id":182893,"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-3933e870d6e631c7"]}],"region_exact":{},"incoming_edges":[{"id":"e-3933e870d6e631c7","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-f56f01f99b47548b","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":"d2974375092156574098eef26e5fdb56c9964d1cb0c8e1ca2e77bf20ab9523a9","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-c856ca922af03076","role":"sequence","name":"Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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: 1,      4;\n  2: 1,     20,        9;\n  3: 1,     84,      126,        16;\n  4: 1,    340,     1323,       480,         25;\n  5: 1,   1364,    12672,     10032,       1375,        36;\n  6: 1,   5460,   117117,    183040,      50050,      3276,       49;\n  7: 1,  21844,  1066338,   3136848,    1537250,    190008,     6860,      64;\n  8: 1,  87380,  9646191,  52085280,   43332575,   9053928,   594762,   13056,    81;\n  9: 1, 349524, 87012324, 850513264, 1164697625, 388340316, 41466740, 1612416, 23085, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 9*x^2*P(n-2,x) + (9*x^2+4*x^3)*d/dx P(n-2,x) + 17*x^3*d^2/dx^2 P(n-2,x) + 4*x^4*d^3/dx^3 P(n-2,x) + (1+4*x)*P(n-1,x) + (3*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 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[9*x^2*P[n - 2, x] + (9*x^2 + 4*x^3)*D[P[n - 2, x], x] + 17*x^3*D[P[n - 2, x], {x, 2}] + 4*x^4*D[P[n - 2, x], {x, 3}] + (1 + 4*x)*P[n - 1, x] + (3*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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] = 9*x^2*v[i - 2] + (9*x^2 + 4*x^3)*deriv(v[i - 2], x) + 17*x^3*deriv(deriv(v[i - 2], x), x) + 4*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*x)*v[i - 1] + (3*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-c856ca922af03076","steps":["(\u03b8+1)\u00b2"],"edges":["e-3933e870d6e631c7"],"text":"A036969 \u2192 (\u03b8+1)\u00b2"},"headline":"A036969 \u2192 (\u03b8+1)\u00b2","chain":null}