{"id":"n-00cc3b258a483581","revision":1,"first_index":0,"fingerprint":"f6-fad61a00601faf8c","name":"theta_plus_one_squared of Triangle, row sums = number of edges in n-dimensional hypercubes.","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","12"],["1","24","45"],["1","36","135","112"],["1","48","270","448","225"],["1","60","450","1120","1125","396"],["1","72","675","2240","3375","2376","637"],["1","84","945","3920","7875","8316","4459","960"],["1","96","1260","6272","15750","22176","17836","7680","1377"],["1","108","1620","9408","28350","49896","53508","34560","12393","1900"],["1","120","2025","13440","47250","99792","133770","115200","61965","19000","2541"],["1","132","2475","18480","74250","182952","294294","316800","227205","104500","27951","3312"],["1","144","2970","24640","111375","313632","588588","760320","681615","418000","167706","39744","4225"],["1","156","3510","32032","160875","509652","1093092","1647360","1772199","1358500","726726","258336","54925","5292"],["1","168","4095","40768","225225","792792","1912911","3294720","4135131","3803800","2543541","1205568","384475","74088","6525"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","70","284","992","3152","9376","26560","72448","191744","495104","1252352","3112960","7622656","18423808"],"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":[["1","-8","3","0"],["0","11","-11","0"],["0","0","11","-1"],["0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["1","4","6","4","1"],["0","11","22","15","4"],["0","0","11","16","5"],["0","0","0","2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163532,"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-e317db6457cebe23"]},{"id":163533,"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-e317db6457cebe23"]},{"id":163534,"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-e317db6457cebe23"]}],"region_exact":{},"incoming_edges":[{"id":"e-e317db6457cebe23","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-00cc3b258a483581","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":"1bbf080b662d82e5048514965987ef01f1d8754eff0c1fac2819a752549f5479","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-6202e291a2627bce","role":"sequence","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","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,  12;\n  2: 1,  24,   45;\n  3: 1,  36,  135,  112;\n  4: 1,  48,  270,  448,   225;\n  5: 1,  60,  450, 1120,  1125,   396;\n  6: 1,  72,  675, 2240,  3375,  2376,   637;\n  7: 1,  84,  945, 3920,  7875,  8316,  4459,   960;\n  8: 1,  96, 1260, 6272, 15750, 22176, 17836,  7680,  1377;\n  9: 1, 108, 1620, 9408, 28350, 49896, 53508, 34560, 12393, 1900;\n","recurrence_sentence":"The row polynomials satisfy (1+(11*n-8)*x+(11*n^2-11*n+3)*x^2+(2*n^3-n^2)*x^3)*P(n,x) = (1+(11*n+4)*x+(11*n^2+22*n+6)*x^2+(2*n^3+16*n^2+15*n+4)*x^3+(2*n^3+5*n^2+4*n+1)*x^4)*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[((1 + (11*n + 4)*x + (11*n^2 + 22*n + 6)*x^2 + (2*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*P[n - 1, x])/(1 + (11*n - 8)*x + (11*n^2 - 11*n + 3)*x^2 + (2*n^3 - n^2)*x^3)]];\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] = ((1 + (11*n + 4)*x + (11*n^2 + 22*n + 6)*x^2 + (2*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*v[i - 1])/(1 + (11*n - 8)*x + (11*n^2 - 11*n + 3)*x^2 + (2*n^3 - n^2)*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-c8c19eba9d68881f","steps":["2\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-c4ea45619bc7f60a","e-e317db6457cebe23"],"text":"A007318 \u2192 2\u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}