{"id":"n-4a61d31ab117b16d","revision":0,"first_index":0,"fingerprint":"f6-e07ab4f767b2303c","name":"theta_plus_one_squared of Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","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","40","9"],["1","364","315","16"],["1","3280","8694","1344","25"],["1","29524","224730","81312","4125","36"],["1","265720","5684679","4383808","461175","10296","49"],["1","2391484","142714845","224912688","44204875","1909908","22295","64"],["1","21523360","3573251964","11274436992","3932020950","294753888","6367452","43520","81"],["1","193710244","89379726660","558799860544","336110004750","41327330616","1477292180","18108672","78489","100"],["1","1743392200","2234929014549","27540090458496","28098035166850","5484605411376","305913467314","6003975680","45602109","133000","121"],["1","15690529804","55877147996175","1353437639603280","2318972239856250","704098425416760","59164533341550","1750454975104","20777791221","104311900","214291","144"],["1","141214768240","1396964003596434","66417781492553920","189951497740236375","88535229500820960","10957162324872540","471128494579200","8220201240735","63308190000","220719730","331200","169"],["1","1270932914164","34924417823139390","3256954784697091296","15489849100541261875","10986292926345276540","1972266717501798900","120315306969949440","2971910159524215","33002653183500","173940310830","437879520","494325","196"],["1","11438396227480","873113305177541619","159652872304065276864","1259766768994931417025","1351646826792557878440","348266640629773973835","29646965903343014400","1011156096486340395","15582982687545000","116640887428065","438641611200","822853395","716184","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","50","696","13344","339728","10805728","416156160","19102398208","1027306112256","63671366621696","4493337014036480","357738188501909504","31863609612002824192","3150880055527771824128"],"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":[["-1","12","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","28","24","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","60","52"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","16","16"]]},"deriv_order":3,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188531,"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-03b2284d4ddf4446"]},{"id":188532,"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-03b2284d4ddf4446"]},{"id":188533,"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-03b2284d4ddf4446"]}],"region_exact":{},"incoming_edges":[{"id":"e-03b2284d4ddf4446","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-4a61d31ab117b16d","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":"d2c6cc21781c953be710dc99e0545b00334a197b34cec4b48fc3f424e30e8da3","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-65e47d0258ed4170","role":"sequence","name":"Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","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,        40,           9;\n  3: 1,       364,         315,           16;\n  4: 1,      3280,        8694,         1344,           25;\n  5: 1,     29524,      224730,        81312,         4125,          36;\n  6: 1,    265720,     5684679,      4383808,       461175,       10296,         49;\n  7: 1,   2391484,   142714845,    224912688,     44204875,     1909908,      22295,       64;\n  8: 1,  21523360,  3573251964,  11274436992,   3932020950,   294753888,    6367452,    43520,    81;\n  9: 1, 193710244, 89379726660, 558799860544, 336110004750, 41327330616, 1477292180, 18108672, 78489, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-12*x-9*x^2)*P(n-2,x) + (28*x+24*x^2+4*x^3)*d/dx P(n-2,x) + (60*x^2+52*x^3)*d^2/dx^2 P(n-2,x) + (16*x^3+16*x^4)*d^3/dx^3 P(n-2,x) + (2+4*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*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[-(1 - 12*x - 9*x^2)*P[n - 2, x] + (28*x + 24*x^2 + 4*x^3)*D[P[n - 2, x], x] + (60*x^2 + 52*x^3)*D[P[n - 2, x], {x, 2}] + (16*x^3 + 16*x^4)*D[P[n - 2, x], {x, 3}] + (2 + 4*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*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] = -(1 - 12*x - 9*x^2)*v[i - 2] + (28*x + 24*x^2 + 4*x^3)*deriv(v[i - 2], x) + (60*x^2 + 52*x^3)*deriv(deriv(v[i - 2], x), x) + (16*x^3 + 16*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2 + 4*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*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-65e47d0258ed4170","steps":["(\u03b8+1)\u00b2"],"edges":["e-03b2284d4ddf4446"],"text":"A160562 \u2192 (\u03b8+1)\u00b2"},"headline":"A160562 \u2192 (\u03b8+1)\u00b2","chain":null}