{"id":"n-846aecace8de9c1d","revision":0,"first_index":0,"fingerprint":"f6-35caaae7884ed898","name":"theta_plus_one_squared of reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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"],["3","8"],["6","36","27"],["10","96","162","64"],["15","200","540","480","125"],["21","360","1350","1920","1125","216"],["28","588","2835","5600","5250","2268","343"],["36","896","5292","13440","17500","12096","4116","512"],["45","1296","9072","28224","47250","45360","24696","6912","729"],["55","1800","14580","53760","110250","136080","102900","46080","10935","1000"],["66","2420","22275","95040","231000","349272","339570","211200","80190","16500","1331"],["78","3168","32670","158400","445500","798336","950796","760320","400950","132000","23958","1728"],["91","4056","46332","251680","804375","1667952","2354352","2306304","1563705","715000","207636","33696","2197"],["105","5096","63882","384384","1376375","3243240","5297292","6150144","5108103","3003000","1211210","314496","46137","2744"],["120","6300","85995","567840","2252250","5945940","11036025","14826240","14594580","10510500","5450445","1965600","461370","61740","3375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","69","332","1360","4992","16912","53888","163584","477440","1348864","3707904","9957376","26206208","67768320"],"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"],["3","8"],["6","36","27"],["10","96","162","64"],["15","200","540","480","125"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","5","10","10","5","1"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["-5","-20","-30","-20","-5"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["10","30","30","10"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-10","-20","-10"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173996,"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-186a584f3a965d35"]},{"id":173997,"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-186a584f3a965d35"]},{"id":173998,"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-186a584f3a965d35"]}],"region_exact":{},"incoming_edges":[{"id":"e-186a584f3a965d35","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-846aecace8de9c1d","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":"cf471b70c7be916453b6bd57d0e00b429eee010109dc75ba161c9c24b9d5bcd7","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-2c11b65ec5b65118","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:  3,    8;\n  2:  6,   36,    27;\n  3: 10,   96,   162,    64;\n  4: 15,  200,   540,   480,    125;\n  5: 21,  360,  1350,  1920,   1125,    216;\n  6: 28,  588,  2835,  5600,   5250,   2268,    343;\n  7: 36,  896,  5292, 13440,  17500,  12096,   4116,   512;\n  8: 45, 1296,  9072, 28224,  47250,  45360,  24696,  6912,   729;\n  9: 55, 1800, 14580, 53760, 110250, 136080, 102900, 46080, 10935, 1000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 6 + 36*x + 27*x^2, P(3,x) = 10 + 96*x + 162*x^2 + 64*x^3, P(4,x) = 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[2, x_] := 6 + 36*x + 27*x^2;\nP[3, x_] := 10 + 96*x + 162*x^2 + 64*x^3;\nP[4, x_] := 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; v[3] = 6 + 36*x + 27*x^2; v[4] = 10 + 96*x + 162*x^2 + 64*x^3; v[5] = 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*x)*v[i - 1]); 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-787407680a56b31c","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-03e6ee68811b0a8b","e-186a584f3a965d35"],"text":"A124932 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A124932 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}