{"id":"n-8aafd380e3a0603f","revision":0,"first_index":0,"fingerprint":"f6-f105a98b26320007","name":"theta_plus_one_squared of shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*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"],["2"],["7","16"],["20","64"],["61","288","144"],["182","1088","864"],["547","4016","4752","1024"],["1640","14208","21888","8192"],["4921","49216","94176","53248","6400"],["14762","167296","381888","286720","64000"],["44287","560976","1489248","1402880","480000","36864"],["132860","1860032","5629824","6381568","2944000","442368"],["398581","6111584","20776176","27561984","16096000","3760128","200704"],["1195742","19929024","75182112","114302976","80908800","25804800","2809856"],["3587227","64570096","267697008","459033600","382368000","155602944","26693632","1048576"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","23","84","493","2134","10339","45928","207961","914666","4014255","17390652","74905157","320133310","1360601083"],"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"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["144","162","25","-44","0"],["0","240","104","-26","0"],["0","0","184","78","4"],["0","0","0","48","8"],["0","0","0","0","4"]]},"terms":[{"coeff":{"coeffs":[["-432","-1782","-2547","-1420","-272","-64"],["720","882","-1039","-1982","-888","-128"],["0","1704","3004","1332","36","-16"],["0","0","1208","1894","788","112"],["0","0","0","288","348","80"],["0","0","0","0","20","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-216","-1188","-2610","-2942","-1848","-672","-128"],["0","-864","-3720","-6292","-5276","-2224","-384"],["0","0","-888","-3116","-3932","-2120","-416"],["0","0","0","-288","-816","-720","-192"],["0","0","0","0","-24","-56","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175218,"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-e1368094e7fe67d7"]},{"id":175219,"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-e1368094e7fe67d7"]},{"id":175220,"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-e1368094e7fe67d7"]}],"region_exact":{},"incoming_edges":[{"id":"e-e1368094e7fe67d7","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-8aafd380e3a0603f","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":"efdda98f434d9e3a2be0e2c510d1c80c148f340aed1ff598cf0595a45d475a4a","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-e5bdf85b655e989a","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*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:     2;\n  2:     7,     16;\n  3:    20,     64;\n  4:    61,    288,    144;\n  5:   182,   1088,    864;\n  6:   547,   4016,   4752,   1024;\n  7:  1640,  14208,  21888,   8192;\n  8:  4921,  49216,  94176,  53248,  6400;\n  9: 14762, 167296, 381888, 286720, 64000;\n","recurrence_sentence":"The row polynomials satisfy (144*n+(240*n^2+162*n)*x+(184*n^3+104*n^2+25*n)*x^2+(48*n^4+78*n^3-26*n^2-44*n)*x^3+(4*n^5+8*n^4+4*n^3)*x^4)*P(n,x) = (720*n-432+(1704*n^2+882*n-1782)*x+(1208*n^3+3004*n^2-1039*n-2547)*x^2+(288*n^4+1894*n^3+1332*n^2-1982*n-1420)*x^3+(20*n^5+348*n^4+788*n^3+36*n^2-888*n-272)*x^4+(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64)*x^5)*P(n-1,x) - (216+(864*n+1188)*x+(888*n^2+3720*n+2610)*x^2+(288*n^3+3116*n^2+6292*n+2942)*x^3+(24*n^4+816*n^3+3932*n^2+5276*n+1848)*x^4+(56*n^4+720*n^3+2120*n^2+2224*n+672)*x^5+(32*n^4+192*n^3+416*n^2+384*n+128)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((720*n - 432 + (1704*n^2 + 882*n - 1782)*x + (1208*n^3 + 3004*n^2 - 1039*n - 2547)*x^2 + (288*n^4 + 1894*n^3 + 1332*n^2 - 1982*n - 1420)*x^3 + (20*n^5 + 348*n^4 + 788*n^3 + 36*n^2 - 888*n - 272)*x^4 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^5)*P[n - 1, x] - (216 + (864*n + 1188)*x + (888*n^2 + 3720*n + 2610)*x^2 + (288*n^3 + 3116*n^2 + 6292*n + 2942)*x^3 + (24*n^4 + 816*n^3 + 3932*n^2 + 5276*n + 1848)*x^4 + (56*n^4 + 720*n^3 + 2120*n^2 + 2224*n + 672)*x^5 + (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^6)*D[P[n - 1, x], x])/(144*n + (240*n^2 + 162*n)*x + (184*n^3 + 104*n^2 + 25*n)*x^2 + (48*n^4 + 78*n^3 - 26*n^2 - 44*n)*x^3 + (4*n^5 + 8*n^4 + 4*n^3)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((720*n - 432 + (1704*n^2 + 882*n - 1782)*x + (1208*n^3 + 3004*n^2 - 1039*n - 2547)*x^2 + (288*n^4 + 1894*n^3 + 1332*n^2 - 1982*n - 1420)*x^3 + (20*n^5 + 348*n^4 + 788*n^3 + 36*n^2 - 888*n - 272)*x^4 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^5)*v[i - 1] - (216 + (864*n + 1188)*x + (888*n^2 + 3720*n + 2610)*x^2 + (288*n^3 + 3116*n^2 + 6292*n + 2942)*x^3 + (24*n^4 + 816*n^3 + 3932*n^2 + 5276*n + 1848)*x^4 + (56*n^4 + 720*n^3 + 2120*n^2 + 2224*n + 672)*x^5 + (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^6)*deriv(v[i - 1], x))/(144*n + (240*n^2 + 162*n)*x + (184*n^3 + 104*n^2 + 25*n)*x^2 + (48*n^4 + 78*n^3 - 26*n^2 - 44*n)*x^3 + (4*n^5 + 8*n^4 + 4*n^3)*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-cf38558d99f4fe37","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-7b35457a4f9e8cf8","e-e1368094e7fe67d7"],"text":"A229032 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A229032 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}