{"id":"n-a3755ba558645334","revision":1,"first_index":0,"fingerprint":"f6-18d84775e29cd07e","name":"derivative of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["8","6"],["18","36","12"],["32","108","96","20"],["50","240","360","200","30"],["72","450","960","900","360","42"],["98","756","2100","2800","1890","588","56"],["128","1176","4032","7000","6720","3528","896","72"],["162","1728","7056","15120","18900","14112","6048","1296","90"],["200","2430","11520","29400","45360","44100","26880","9720","1800","110"],["242","3300","17820","52800","97020","116424","92400","47520","14850","2420","132"],["288","4356","26400","89100","190080","271656","266112","178200","79200","21780","3168","156"],["338","5616","37752","143000","347490","576576","672672","555984","321750","125840","30888","4056","182"],["392","7098","52416","220220","600600","1135134","1537536","1513512","1081080","550550","192192","42588","5096","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","14","66","256","880","2784","8288","23552","64512","171520","444928","1130496","2822144","6938624"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"],["8","6"],["18","36","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["-12"],["-14"],["12"],["7"],["-6"],["1"]]},"terms":[{"coeff":{"coeffs":[["24","72","72","24"],["0","0","0","0"],["-8","-24","-24","-8"],["15","45","45","15"],["-5","-15","-15","-5"],["-3","-9","-9","-3"],["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["12","48","72","48","12"],["6","24","36","24","6"],["-6","-24","-36","-24","-6"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["36","72","36"],["6","12","6"],["-60","-120","-60"],["9","18","9"],["12","24","12"],["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-24","-72","-72","-24"],["-24","-72","-72","-24"],["12","36","36","12"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-72","-72"],["-30","-30"],["63","63"],["3","3"],["-15","-15"],["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["18","36","18"],["-6","-12","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163895,"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-98dce694e357661a"]},{"id":163896,"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-98dce694e357661a"]},{"id":163897,"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-98dce694e357661a"]}],"region_exact":{},"incoming_edges":[{"id":"e-98dce694e357661a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-a3755ba558645334","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"f3f579ba7185793218a45d59350eb50d2e92c962832bcd3b6dd1f30d85c14209","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-14d5814e0f152cf3","role":"sequence","name":"Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(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:   0;\n  1:   2;\n  2:   8,    6;\n  3:  18,   36,   12;\n  4:  32,  108,   96,    20;\n  5:  50,  240,  360,   200,    30;\n  6:  72,  450,  960,   900,   360,    42;\n  7:  98,  756, 2100,  2800,  1890,   588,   56;\n  8: 128, 1176, 4032,  7000,  6720,  3528,  896,   72;\n  9: 162, 1728, 7056, 15120, 18900, 14112, 6048, 1296, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^6-6*n^5+7*n^4+12*n^3-14*n^2-12*n)*P(n,x) = (n^6-3*n^5-5*n^4+15*n^3-8*n^2+24+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x^2+(n^6-3*n^5-5*n^4+15*n^3-8*n^2+24)*x^3)*P(n-3,x) - (6*n^2-6*n-12+(24*n^2-24*n-48)*x+(36*n^2-36*n-72)*x^2+(24*n^2-24*n-48)*x^3+(6*n^2-6*n-12)*x^4)*d/dx P(n-3,x) - (3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n+(6*n^6-24*n^5-18*n^4+120*n^3-12*n^2-72*n)*x+(3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n)*x^2)*P(n-2,x) + (12*n^2-24*n-24+(36*n^2-72*n-72)*x+(36*n^2-72*n-72)*x^2+(12*n^2-24*n-24)*x^3)*d/dx P(n-2,x) + (3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n+(3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n)*x)*P(n-1,x) - (6*n^2-18*n+(12*n^2-36*n)*x+(6*n^2-18*n)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 8 + 6*x, P(3,x) = 18 + 36*x + 12*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 8 + 6*x;\nP[3, x_] := 18 + 36*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*P[n - 3, x] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*D[P[n - 3, x], x] - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*P[n - 2, x] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*D[P[n - 2, x], x] + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*P[n - 1, x] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*D[P[n - 1, x], x])/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 8 + 6*x; v[4] = 18 + 36*x + 12*x^2; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*v[i - 3] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*deriv(v[i - 3], x) - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*v[i - 2] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*deriv(v[i - 2], x) + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*v[i - 1] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*deriv(v[i - 1], x))/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)); 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":["\u2212D\u00b2+(\u22121+n)D+1+n","D"],"edges":["e-168270346f6f7c93","e-98dce694e357661a"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 D"},"headline":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 D","chain":null}