{"id":"n-6a3feb59f46c915f","revision":0,"first_index":1,"fingerprint":"f6-cbb0d2a2a7c53ac6","name":"theta_plus_one of reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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","2"],["1","8","3"],["1","20","27","4"],["1","40","126","64","5"],["1","70","420","480","125","6"],["1","112","1134","2400","1375","216","7"],["1","168","2646","9240","9625","3276","343","8"],["1","240","5544","29568","50050","30576","6860","512","9"],["1","330","10692","82368","210210","206388","82320","13056","729","10"],["1","440","19305","205920","750750","1100736","699720","195840","23085","1000","11"],["1","572","33033","471900","2359500","4900896","4618152","2046528","423225","38500","1331","12"],["1","728","54054","1006720","6685250","18903456","25069968","16372224","5332635","847000","61226","1728","13"],["1","910","85176","2022592","17381650","64849356","116396280","106419456","50837787","12662650","1591876","93600","2197","14"],["1","1120","129948","3861312","42031990","201753552","475284810","585307008","389756367","141821680","27857830","2839200","138411","2744","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","52","236","1102","5245","25307","123360","606104","2996808","14893650","74335003","372343545","1870785988"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["-4","0"],["-4","-2"],["8","2"],["0","6"],["0","-10"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-4","16","4"],["-4","-46","-14"],["8","30","14"],["0","6","5"],["0","-10","-16"],["0","4","7"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-24","12","6"],["0","36","-34","-18"],["0","24","20","12"],["0","0","-22","14"],["0","0","-6","-16"],["0","0","12","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16","-4","2"],["0","0","40","14","-6"],["0","0","0","-16","4"],["0","0","0","-20","5"],["0","0","0","20","-5"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175035,"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-74dea4565e90cda0"]},{"id":175036,"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-74dea4565e90cda0"]},{"id":175037,"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-74dea4565e90cda0"]}],"region_exact":{},"incoming_edges":[{"id":"e-74dea4565e90cda0","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-6a3feb59f46c915f","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":"ad3ce858a5203ff5e9a3de400745d5d36a3f92f8fb77c05ccdf4c992c025c393","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-3aeb7159e408f97a","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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,   2;\n   3: 1,   8,     3;\n   4: 1,  20,    27,     4;\n   5: 1,  40,   126,    64,      5;\n   6: 1,  70,   420,   480,    125,      6;\n   7: 1, 112,  1134,  2400,   1375,    216,     7;\n   8: 1, 168,  2646,  9240,   9625,   3276,   343,     8;\n   9: 1, 240,  5544, 29568,  50050,  30576,  6860,   512,   9;\n  10: 1, 330, 10692, 82368, 210210, 206388, 82320, 13056, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-4*n^2-4*n+(4*n^6-10*n^5+6*n^4+2*n^3-2*n^2)*x)*P(n,x) = (8*n^3-4*n^2-4*n+(4*n^6-10*n^5+6*n^4+30*n^3-46*n^2+16*n)*x+(7*n^6-16*n^5+5*n^4+14*n^3-14*n^2+4*n)*x^2)*P(n-1,x) + ((24*n^2+36*n-24)*x+(12*n^5-6*n^4-22*n^3+20*n^2-34*n+12)*x^2+(2*n^5-16*n^4+14*n^3+12*n^2-18*n+6)*x^3)*d/dx P(n-1,x) + ((40*n-16)*x^2+(20*n^4-20*n^3-16*n^2+14*n-4)*x^3-(5*n^4-5*n^3-4*n^2+6*n-2)*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 30*n^3 - 46*n^2 + 16*n)*x + (7*n^6 - 16*n^5 + 5*n^4 + 14*n^3 - 14*n^2 + 4*n)*x^2)*P[n - 1, x] + ((24*n^2 + 36*n - 24)*x + (12*n^5 - 6*n^4 - 22*n^3 + 20*n^2 - 34*n + 12)*x^2 + (2*n^5 - 16*n^4 + 14*n^3 + 12*n^2 - 18*n + 6)*x^3)*D[P[n - 1, x], x] + ((40*n - 16)*x^2 + (20*n^4 - 20*n^3 - 16*n^2 + 14*n - 4)*x^3 - (5*n^4 - 5*n^3 - 4*n^2 + 6*n - 2)*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 2*n^3 - 2*n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 30*n^3 - 46*n^2 + 16*n)*x + (7*n^6 - 16*n^5 + 5*n^4 + 14*n^3 - 14*n^2 + 4*n)*x^2)*v[i - 1] + ((24*n^2 + 36*n - 24)*x + (12*n^5 - 6*n^4 - 22*n^3 + 20*n^2 - 34*n + 12)*x^2 + (2*n^5 - 16*n^4 + 14*n^3 + 12*n^2 - 18*n + 6)*x^3)*deriv(v[i - 1], x) + ((40*n - 16)*x^2 + (20*n^4 - 20*n^3 - 16*n^2 + 14*n - 4)*x^3 - (5*n^4 - 5*n^3 - 4*n^2 + 6*n - 2)*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 2*n^3 - 2*n^2)*x)); 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-5c0c5bce3aee8449","steps":["reverse","\u03b8+1"],"edges":["e-5d51438768cc4177","e-74dea4565e90cda0"],"text":"A391045 \u2192 reverse \u2192 \u03b8+1"},"headline":"A391045 \u2192 reverse \u2192 \u03b8+1","chain":null}