{"id":"n-abdfd8edac4c2935","revision":0,"first_index":0,"fingerprint":"f6-5e04371f1947d221","name":"theta_plus_one of shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","4"],["9","16"],["39","108","48"],["189","648","408"],["1107","4740","4608","1088"],["7281","36576","44928","15872"],["54351","316764","486432","264192","39680"],["448821","2961144","5377752","3750912","884480"],["4085883","30180276","63754128","55710528","20171520","2122752"],["40533129","330942672","794559456","826419456","389370240","67104768"],["435847959","3900302604","10508155920","12746661120","7544582400","1998208512","156577792"],["5045745069","49093525224","146558830680","202732646400","143588563200","50030134272","6663150592"],["62594829027","658013925348","2156905768032","3355357608000","2778856070400","1220018789376","250263681024","15230058496"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","25","195","1245","11543","104657","1161419","13423109","176025087","2448929721","37290336307","603712595437","10497240729703"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","-3","0","0"],["0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["-54","-144","-60","96","64","0"],["36","60","12","32","64","16"],["0","12","0","-36","-16","8"],["0","0","0","-8","-16","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-27","-99","-102","18","80","32","0"],["0","-18","-42","12","96","76","16"],["0","0","0","12","40","44","16"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-18","-9","24","8","0"],["0","-12","0","10","-8"],["0","0","0","8","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","9","21","12","0","0"],["0","0","0","-6","-14","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178770,"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-36c36d82c85a8163"]},{"id":178771,"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-36c36d82c85a8163"]},{"id":178772,"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-36c36d82c85a8163"]}],"region_exact":{},"incoming_edges":[{"id":"e-36c36d82c85a8163","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-abdfd8edac4c2935","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":"df639730b5d2c0cb0b499dcdaeebf82e7144cf4a7bbb20884fa7e46697561a58","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-e8d6574c8023ca59","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-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;\n  2:      3,       4;\n  3:      9,      16;\n  4:     39,     108,      48;\n  5:    189,     648,     408;\n  6:   1107,    4740,    4608,    1088;\n  7:   7281,   36576,   44928,   15872;\n  8:  54351,  316764,  486432,  264192,  39680;\n  9: 448821, 2961144, 5377752, 3750912, 884480;\n","recurrence_sentence":"The row polynomials satisfy -(3*x-2*n*x^3)*P(n,x) = (36*n-54+(12*n^2+60*n-144)*x+(12*n-60)*x^2-(8*n^3+36*n^2-32*n-96)*x^3-(16*n^3+16*n^2-64*n-64)*x^4-(8*n^3-8*n^2-16*n)*x^5)*P(n-2,x) - (27+(18*n+99)*x+(42*n+102)*x^2-(12*n^2+12*n+18)*x^3-(40*n^2+96*n+80)*x^4-(44*n^2+76*n+32)*x^5-(16*n^2+16*n)*x^6)*d/dx P(n-2,x) - (18+(12*n+9)*x-24*x^2-(8*n^2+10*n+8)*x^3-(8*n^2-8*n)*x^4)*P(n-1,x) + (9*x+21*x^2-(6*n-12)*x^3-14*n*x^4-8*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((36*n - 54 + (12*n^2 + 60*n - 144)*x + (12*n - 60)*x^2 - (8*n^3 + 36*n^2 - 32*n - 96)*x^3 - (16*n^3 + 16*n^2 - 64*n - 64)*x^4 - (8*n^3 - 8*n^2 - 16*n)*x^5)*P[n - 2, x] - (27 + (18*n + 99)*x + (42*n + 102)*x^2 - (12*n^2 + 12*n + 18)*x^3 - (40*n^2 + 96*n + 80)*x^4 - (44*n^2 + 76*n + 32)*x^5 - (16*n^2 + 16*n)*x^6)*D[P[n - 2, x], x] - (18 + (12*n + 9)*x - 24*x^2 - (8*n^2 + 10*n + 8)*x^3 - (8*n^2 - 8*n)*x^4)*P[n - 1, x] + (9*x + 21*x^2 - (6*n - 12)*x^3 - 14*n*x^4 - 8*n*x^5)*D[P[n - 1, x], x])/(-3*x + 2*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((36*n - 54 + (12*n^2 + 60*n - 144)*x + (12*n - 60)*x^2 - (8*n^3 + 36*n^2 - 32*n - 96)*x^3 - (16*n^3 + 16*n^2 - 64*n - 64)*x^4 - (8*n^3 - 8*n^2 - 16*n)*x^5)*v[i - 2] - (27 + (18*n + 99)*x + (42*n + 102)*x^2 - (12*n^2 + 12*n + 18)*x^3 - (40*n^2 + 96*n + 80)*x^4 - (44*n^2 + 76*n + 32)*x^5 - (16*n^2 + 16*n)*x^6)*deriv(v[i - 2], x) - (18 + (12*n + 9)*x - 24*x^2 - (8*n^2 + 10*n + 8)*x^3 - (8*n^2 - 8*n)*x^4)*v[i - 1] + (9*x + 21*x^2 - (6*n - 12)*x^3 - 14*n*x^4 - 8*n*x^5)*deriv(v[i - 1], x))/(-3*x + 2*n*x^3)); 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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","x \u21a6 x+1","\u03b8+1"],"edges":["e-7e5061201b6d9b87","e-33f9ec87155c6d6e","e-36c36d82c85a8163"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}